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SigMath User Guide

1. Font family Sample text Courier Courier Courier Bold Courier bold Courier Oblique Courier oblique Courier BoldOblique Courier bold oblique Times Roman Times Roman Times Bold Times Roman bold Times Italic Times Roman italic Times BoldItalic Times Roman bold italic Helvetica Helvetica Helvetica Bold Helvetica bold Helvetica Oblique Helvetica oblique Helvetica BoldOblique Helvetica bold oblique Symbol symbols ZapfDingbats YY X Tbl 5 3 Font families for the graphic export Encoding This option only takes effect for EPS or PS There are two states default and german Using german the character table is enlarged to display German umlauts and 8 correctly Options Here some further options can be enabled or disabled A list is contained in Table 5 4 Title For POSTSCRIPT files one can add a title Producer For POSTSCRIPT files one can specify the producer to be stored in the target file After treating all options of the export dialog we want to illustrate some subtilities It might be necessary to hide interactive elements that are needed to adjust the figure and that should not be visible in the exported figure any more This is realized by using an virtual env environ ments which is not executed on export Code beginf env EnvBreakIf g1 Get g1 RENDER_MODE gl RENDER text and figure that are not to be exported end fenv This code is also available as a templa
2. e 19 2 1 Environments in SigMath a hee oa E e ee a 22 2 2 Font scaling commands i hani ADe a a e a ee 25 2 3 Shorthands for horizontal spacing o 27 JA Commands for colors s se co ak gaiak aed By ao Bee ae Fo ea tae ees 33 2 5 Symbols in SisMath 24 54 4224 eta abe be ee ee eS 38 3 1 Messages for input arrays 2 a 44 3 2 WINDOWS system messages 2 e 46 3 3 Functions returning ASCII code of special keys o a 46 3 4 MOUSE CUESOL nic a Se ee ES LO A As 48 4 1 Plot properties maca areca a a Fk we a a A aa 64 5 1 Menu items of the graphics export dialog window 66 5 2 Supported file formats in the graphics export o e 67 5 3 Font families for the graphic exporb o e o 68 5 4 Advanced options in graphics export e 69 6 1 Classes in SigeMath tundra A e e de e aa ia a 72 6 2 Constants sr ba A a ee ds A 74 6 3 Data types in SigMath e sa o a cco eccosc es 75 6 4 Reserved words in SigMath 0 0 02 00 00 eee ee 76 6 5 Ranges for variables of type Integer Real and Complex 78 6 6 Operators In SIS Math sie fu a a lo ha i a ee ee as 81 6 7 Automatic type conversion for arithmetic operators 0004 81 6 8 Truth tables conspira a a a a A AAA ad 83 6 9 Determination of the base for Integer conversion 92 6 10 Available array types in SigMath o
3. Example 8 1 If we want to use a numerical routine dft computing a discrete Fourier transform that is contained in the library Numerics d11 we write syst LoadLibrary DocDir Numerics d1l num num dft x Once a library is imported it can be used in every document As soon as the calling docu ment is closed also the library is removed Therefore we remind of the different behavior of the commands GoToPage ShowDoc and ShowModal The first one closes the document while the other two leave the document open Functions in DLLs may have arguments or return values of type Pointer Those can be used to execute other functions contained in different libraries To obtain the correct pointer to a function we use syst FunctionPointer Syntax FunctionPointer libName functionName 151 8 Using and Writing DLLs This command returns the pointer to the function functionName which is contained in a DLL that was previously imported under the identifier libName If no such function exists or the DLL has not been loaded the return value points to the zero address Example 8 2 Let us have a look at the usage of FunctionPointer The DLL DLL1 d11 contains the following function double Test double x return exp x x and provides the function GetTest_VP int argTypel int argSize void argPointer check if function was called with the correct arguments if argType DLL_ARG_VOID_POINTER argType
4. Syntax AxisMaxRange axsNr min max 62 4 2 Three dimensional plots After issuing this command horizontal or vertical scrolling only changes the axis range of axis axsNr within the minimal value min and maximal value max Additionally one can fix the displayed axis ranges by AxisRange 4 1 6 Text If we want to add text to a plot we use the postext environment see Section 2 5 The coordi nates of the corresponding TextPos command are interpreted with respect to the plot coordinate system i e with respect to the not necessarily visible coordinate axes If we want the system to interpret the coordinates with respect to the plot window relative to width and height of the plot we set the property TextPosPlotRect to true Then the lower left corner has coordinates 0 0 the top right corner 1 1 Another important topic is font selection for text and axis labels Especially if one wants to export a SigMath generated plot one has to take care The position of a text block is computed using so called font metrics i e with respect to the chosen font family font size and font shape SigMath is using its own font metrics which might differ from the ones used in e g ADOBE READER If one opens exported graphics in an external viewer application the text is positioned at the location used by SigMath but maybe using different font metrics This can result in shifts or overlapping To prevent such effects we suggest to use mono
5. e 200202020000 93 6 11 Container types in SigMath e 103 GA Types of primitives e pressi dl a eo ee a a ce 123 7 2 provided functions bidid e a a ee 150 8 1 Possible Suffices for DLL functions and properties 153 1 Introduction In this section we want to introduce the basic functionalities of SigMath and show the first steps to create a SigMath document In later sections we give more details on the different functions of SigMath and provide examples to nearly all features Most of the examples are also available as files than can be loaded and tested in SigMath The filenames are indicated in brackets 1 1 What is SigMath SigMath is an application which allows to visualize interactive content in an easy way SigMath works as interpreter of its own programming language and combines the following functional ity e graphics and animation e text layout on the screen e user interaction e programming techniques from basic commands to complex data types and threading Interactive functions allow user interaction and navigation in SigMath documents as known from web pages SigMath s graphical engine is based on OpenGL Each document interpreted by SigMath as an animation i e the content of the document is permanently updated and redrawn This allows to perform user driven mathematical computations and visualize its results in real time The text layout syntax is similar to TFX syn
6. Nui IsKeyDown code It returns true if and only if the key having ASCII code code is pressed ASCII code maps characters we refer to Table 3 3 46 Identifier Description VK_CONTROL CTRL VK_UP Arrow up VK_DOWN Arrow down VK_LEFT Arrow left VK_RIGHT Arrow right VK_ESCAPE ESC VK_PRIOR PageUp VK_NEXT PageDown VK_SHIFT Shift VK_SPACE Space VK_F1 function key F1 Tbl 3 3 Functions returning ASCII code of special keys The code of the other function keys Fi can be obtained by adding i 1 to the value of VK_F1 3 4 Event handling Example 3 3 Imagine we want to allow change of a value in an input field by arrow keys in addition to mouse action or keyboard input Typically we initialize the variable connected to the input field by a default value and provide information on maximal and minimal values and on the step size Using the arrow keys the value of the variable shall be increased or decreased by the value of the step size depending on whether we use Arrow up or Arrow down To enable this behavior for an activated input field we have to add the following lines of code Code Xvariable X 0 0 function ProcessMessage message argl arg2 if message ui WM_KEYUP switch arg1 case ui VK_UP X InputDelta return case ui VK_DOWN X InputDelta return begin document InputDelta 0 1 X Input X OnWinMessage ProcessMessage e
7. Tbl 1 1 Menu items in the editor edly It starts known to ATEX users with the command begin document and ends with the command end document Thus the simplest SigMath documents consists of the following two lines begin document end document If we compile this document SigMath will show a black screen To add more content to this document we can place arbitrary text in between the two command lines Since we have not treated commands yet this text may consist of ordinary characters like literals numbers and punctuation marks Just take the classic Hello World Since text is not bound to any position in SigMath by default it can be moved using the arrow keys upwards downwards to the left and to the right But not only the text is moved but actually the whole page moves accordingly With the following lines of code the text line Hello World is initially shown in white letters on the top left corner of a black screen 16 1 6 SigMath documents Shortcut Action CTRL A Change view between current and last viewed document CTRL X Cut text CTRL C Copy text CTRL V Paste text CTRL F Save and compile the current document CTRL P Declare current document as master document of a larger project CTRL S Save the document CTRL Z Undo the last edit DEL Delete the symbol after the cursor End Jump to the end of the current line CTRL End Jump to the last
8. This chapter deals with exporting graphics generated in SigMath to other file formats The necessary functions are based on the GL2PS package by Christophe Geuzaine To export fig ures little programming knowledge is necessary since SigMath provides a graphical user interface GUI for that task SigMath allows to export to various typical file formats such as e g PDF POSTSCRIPT One can hide or uncover certain parts of the figure If the figure is generated from an animation or can be modified by the user the figure can be polished before exporting it This way SigMath allows to generate high quality figures to be used in other applications Problems might occur in the precise positioning of text Text might slip or be squeezed due to different letter sizes In such cases one has to adjust the positioning in SigMath based on the later file format In order to add the export GUI to a document one adds the following two lines to the preamble of the document variable ZS_FormatingDir ProgDir Shared GraficsExport Formating Include ZS_FormatingDir Preamble txt Here ZS_FormatingDir contains the path to the directory where all necessary applications for graphics export are located The second line includes the file Preamble txt which contains further functions and variables that are working in the background Next add the following line to the page of the document that should be exported Include ZS_FormatingDir Param
9. 0 argType DLL_ARG_NONE return ERROR_ARGUMENT void argPointer Test return DLL_RESULT_OK that returns the pointer to the function Test Another DLL DLL2 d11 provides the function int UseTest_V int argType int argSize void argPointer check if function was called with the correct arguments if argType DLL_ARG_VOID_POINTER argType 0 argType DLL_ARG_NONE return ERROR_ARGUMENT typedef double DtoD double DtoD f DtoD void argPointer double x 0 0 return DLL_RESULT_OK that requires a pointer argPointer to a function of special type Now we write in our SigMath document syst LoadLibrary DocDir DLL1 d11 firstDLL syst LoadLibrary DocDir DLL2 d11 secondDLL variable Test firstDLL GetTest secondDLL UseTest Test 152 8 2 Writing DLLs Then the function Test will be executed 8 2 Writing DLLs In the following we briefly want to describe criteria to write customized DLLs for usage within SigMath On request the files ZS_ArgTypesDLL h ZS_ArgUtDLL h and ZS_ArgUtComp1DLL h are available containing definitions and auxiliary functions A function written in C C has to match the following requirements in order to be ex ported to SigMath Syntax int PrefixDLLIndentifier_ReturnType int argTypel 1 int argSize void argPointer Analogously constants or properties must have the fo
10. 2 9 Tables environment they are contained in Positioning in tables Text alignment in columns is controlled by the ColAlign property Syntax XColAlign pos ColAlign i pos In both cases pos is a Real number Default value is 1 0 corresponding to right aligned text Left alignment is given by 0 0 and centering by 0 5 Other values are admissible especially numbers larger than 1 0 or smaller than 0 0 The second alternative changes the property only for the i th column where enumeration starts with 1 The horizontal positioning of entries is given by the formula beginning of column pos x column width width of entry Thus the parameter pos does not take effect on those entries of the column that determine the width of the column Since there are no restrictions on the number of columns or on their width tables with few columns might appear thin To widen them it add horizontal spacing by writing e g or quad This increases text width and thus increases column width Displaying matrices and vectors To display the values of array data types on the screen we use the command ArrayBlock see p 97 in Section 6 3 2 for further description and examples In this way several arrays can be written in block notation Typically matrices and vectors are written in brackets SigMath uses the command Brackets to perform this task Experienced SigMath users may customize this behavior by modifying the definition in the
11. 35 Quote 85 Ratio 72 Real 75 78 81 82 RECT 55 57 Remove0s 90 rendering 122 Replace 87 Resize 97 return 113 reverse 98 99 rotation 136 russian 24 Save 88 97 SaveToClipBoard 89 SaveToKey 88 Scaling 135 scenery 138 ScreenHeight 72 Screen Width 72 scriptsize 25 SelectedDo 41 selectfont 23 25 SelObj 48 49 SelObjInd 49 50 SelObjMessage 49 Set 102 103 SetBlock 96 SetMouseXY 48 Set Version 95 165 Index Size 85 size 95 104 small 25 SquaredLength 101 String 75 79 84 85 89 string 84 85 89 113 SubString 86 SubstringBefore 86 Sum 99 swap 105 switch 76 107 symbol 24 syst FunctionPointer 151 syst LoadLibrary 151 t 86 TexImage2D 144 TextDepth 29 TextEditor 75 TextPos 28 63 TextPosPixel 28 TextPosPlotRect 63 texture mapping 141 TextureFromString 146 TextureLoad 142 textures 141 Text Width 23 thread BeginThread 114 thread EnterCriticalSection 115 thread InitializeCriticalSection 115 thread LeaveCriticalSection 115 TickLengthFactor 61 64 Time 73 tiny 25 Tolnt 91 92 ToLower 87 TopMargin 23 ToReal 91 92 ToString 89 97 ToUpper 87 transformation 135 translations 136 Transpose 98 166 Trim 88 true 76 78 TrueString 91 TypeOf 76 underline 25 use_class 76 use_class className 71 ValueOf 77 94 100 variable 75 76 Version 95 vertex 122 Vertex 122 VK_CONTROL 46 V
12. For conversion to floating point numbers we use the ToReal command instead Syntax string ToReal s string ToReal s out The effect is similar to ToInt The first variant returns the number the second writes it to the variable out and returns true on successful conversion This time s has to be of the form w s ddd ddd fmt s ddd where w marks whitespace s the sign or ddd the digits and fmt stands for the exponential sign e or E Of course stands for the decimal dot All elements are optional Additional values for s are INF and INF for plus and minus infinity see Section 3 2 and NAN for not a number 6 3 2 Arrays Arrays can be used to store several variables of the same data type in one place Table 6 10 contains a list of data types that allow grouping within an array Arrays constitute the SigMath counterpart to mathematical objects like vectors or matrices Definition To define an array we specify the name of the variable the data type of the elements to store in it its dimensionality and sizes in its dimensions The general syntax reads as follows 92 6 3 Complex data types Array type Data type of the elements RealArray Real IntegerArray Integer ComplexArray Complex StringArray String BoolArray Bool Tbl 6 10 Available array types in SigMath Syntax variable ArrayName ArrayType Number_Elements_1 Number_E
13. In addition there is the OpenGL Utility Library short glu which allows to generate more complex graphical elements such as balls or cylinders see Section 7 2 8 It is contained in the OpenGL package Thus SigMath s graphics framework eases the task of generating graphics Let us describe the way OpenGL works OpenGL is a finite state machine i e the way a scene is presented is controlled by parameters of the current states As examples may serve the material or position of an object or if the light is turned on or off Most OpenGL commands change states If an object is moved not the whole scenery has to be redrawn Only states are updated and the representation of the scene is recalculated 7 2 Primitive In computer graphics elementary graphical objects are called primitives Elementary objects are vertices lines or polygons As already mentioned in the introduction we can build complex 121 7 Computer graphics graphical objects from primitives We start with vertices and learn how to generate primitives in SigMath We list some predefined functions and glu commands to generate geometrical figures We close with a elaborate example that generates a three dimensional scenery 7 2 1 Vertices A vertex is an invisible point in a three dimensional coordinate system Here point means the mathematical object i e it has no size An object in OpenGL is described by a set of vertices i e the way it looks and the position of
14. sets the value of the left operand which has to be a variable to the value of the right operand If both operands are of the same data type the as signment is directly executed Otherwise SigMath tries to convert the value of the right operand to the data type of the second variant Type conversions of the left operand is only possible during initialization of a variable Thus initialization determines the data type of the variable and acts as an assignment Besides the simple assignment operator Table 6 6 contains the compound assignments and This operators combine assignments with arithmetical operators The new value of the variable on the left is given by its old value combined with the arithmetical operator and the value of the expression on the right hand side For example after Xx 2 the value of the variable x is the twofold of its former value This allows a compact notation of source code and improves readability of the code Assignments operators are also available for complex data types such as arrays We return to that in Section 6 3 2 Incremental operators The increment operator raises the value of the variable by 1 It is an unary operator and is written in front of the name of the variable Similar the decrement operator lowers the value of the variable by 1 Mind that the variable has to be of type Integer or Real Variables of type Complex are not allowed Those operators are especi
15. the line pattern can be designed on a length of 16 pixels 59 4 Plots in SigMath a PlotStyle FILL b PlotLineStipple 1 1111111111110000 Fig 4 3 PlotStyle options We can stretch this pattern using the lineStippleFactor One typical pattern is given by 0000000011111111 which generates 8 empty pixels and 8 pixels of line The same line would have been generated with lineStippleFactor 8 and 0101010101010101 If we want to return to the default behavior we write PlotLineStipple before using the next Plot command If we choose POINT we can specify the size of the points in pixels This is done by assigning a positive integer to PlotPointSize For example on PlotPointSize 3 the point is drawn as a square of size 3x3 pixels Syntax PlotPointSize If several plots are contained on one page we can use the PlotDepth property to specify which objects are near to the observer and hence overlap the ones which are far away The lower the value of PlotDepth the closer the object to the observer Syntax PlotDepth By default the value is 0 i e if we want to add a plot that shall be shown in front of all other objects without specified depth we have to set PlotDepth to e g 1 As we already learned in the introduction we can change the background of a plot as we have done it with the screen background For this we need the two properties PlotBgStyle and PlotBgColor The syntax is the following Syntax PlotB
16. with full qualified path name Without giving the optional argument leadingString the method writes the value of the vari able S to the file It is written in the file directly following the first line beginning with the 88 6 3 Complex data types string key With the optional argument leadingString SigMath searches in the file for the first line beginning with the expression leadingString key If the file does not contain a line beginning with key or leadingString key SigMath appends a line with key or leadingString key followed by the value of the String variable S The method returns a Boolean value true if filename exists and it contains a line with key and false otherwise The method LoadFromKey searches in the file with full qualified path name filename for the first line either beginning with key or leadingString key Then SigMath writes ev erything following this line to S until it detects or the file ends If filename SigMath ignores leadingString and searches the current document for an environment of the following form begin key some text end key If the search is successful SigMath writes the content of this environment to S Additionally LoadFromKey returns true if filename exists and it contains key and false otherwise If we want to exchange text data via the clipboard we can use the methods LoadFromClip board and SaveToClipBoard Syntax S LoadFromClipboard S SaveToCl
17. 1 6 TickLengthFactor PlotTick 1 10 end plot2d end document Il l A l N Finally all objects like axes and labels have their own depth which can be modified by Axes Depth see TextDepth in Section 2 5 If we want to add additional objects or text to mark certain positions at the text we can also use the AxisPos command to detect the position of the axis on the screen Syntax AxisPos axisNr 61 4 Plots in SigMath il 0 9 8 7 6 5 4 3 2 1 0 1 Fig 4 5 Placing tick marks manually It returns the second coordinate of the first axis if axisNr 1 and the first coordinate of the second axis if axisNr 2 For other values or if no plot object is active an error message is returned 4 1 5 Interactive plots SigMath s overall design principle is animation and interaction Thus plot objects can be mod ified interactively too To enable this one has to set the property Interactive to true Syntax Interactive true This allows the user to move the plot window on the screen by pressing the left mouse button and moving the mouse Horizontal movement of the mouse amounts to horizontal scrolling of the visible x axis area and leads to automatic computation of the corresponding vertical axis area By vertical movements one can zoom in or out yielding an interactive stretching or squeezing of the part of the x axis To limit the effect of interactive modifications one can use AxisMaxRange
18. 5 1 0 TextPos 0 5 0 5 1 5 2 Fig 2 3 Positioning of text using TextPos For the positioning of text within plot environments we refer to Section 4 1 6 p 63 Though the text on the screen is displayed in two dimensions there exists a third coordinate called depth It can be adjusted by changing the value of the property TextDepth Syntax TextDepth The default value is 0 Depth is used if SigMath has to decide which of two overlapping objects appears closer to observer and hence hides the other Here closer means that the depth of the object in the foreground is lower than the depth of the object in the background If TextDepth exceeds 1000 1000 the text is not shown at all 2 6 Enumerations Enumerations are numbered listings used to sort text They can be generated by using the enumerate environment Nesting enumerations is possible List items are defined by using the item command To split longer text into several lines one has to use command Following 29 2 Word processing lines belonging to the same item are indented as the first line To generate subitems we start a new sub list by the command nextlevel followed by item commands To end the sublist we write prevlevel and resume to the previous enumeration level Now new items can be added The syntax for an enumerations thus looks as follows Syntax begin enumerate item item 1 of level 0 item item 2 of level OM second line of the second
19. 5 4 Advanced options in graphics export After exporting a SigMath document test txt two files are generated Besides the ex ported graphics file e g test eps a file test options containing all options used to generate test eps can be found in the corresponding directory If one opens test txt in SigMath again all options are automatically loaded from test options To save one frame in a running animation we stop the animation by pressing CTRL Space and open the Export dialog for the static figure by CTRL S 69 6 Programming This chapter treats the programming language of SigMath We introduce all fundamental struc tures in SigMath Section 6 2 deals with properties and methods of all implemented data types Section 6 4 contains information on control structures and Section 6 5 deals with functions We close this chapter with a section on debugging tools in SigMath 6 1 Classes Commands in SigMath are grouped within certain classes To tell SigMath s interpreter where it can find the definition of the used function we have to provide the name of the class the command is contained Then the interpreter looks up the definition of the command and checks whether all formal requirements for execution of this command are met i e if all required argu ments are specified or all arguments are of correct type We give a list of all classes in SigMath in Table 6 1 The most important classes and commands are then described in more
20. 6 8 828 true false true false A lA true true false true true true true false false false false false true false false true Tbl 6 8 Truth tables Bitwise operators Bitwise operators allow the direct manipulation of the coding of Integer variables Using AND OR and NOT operators data elements of type Integer may be composed or negated We use the same symbols as for the logical operators amp amp and A bit with entry 1 is then interpreted as true and with entry O as false Integers in SigMath always use 32 digits thus every bit operator results in 32 comparisons or negations The operator is performed bitwise i 810 0 010002 j 510 0 001012 i amp amp j Oi0 0 000002 i I j 1310 0 011012 li ToS 910 1 10111 gt 13 610 1 110102 Besides those there are two more bit operators which are given by the functions BinaryShift and BinaryRotation They can be used to shift the bit pattern of an Integer variable to the left or to the right and thus change the value of the variable Let us have a look at the syntax and a small example Syntax BinaryShift i num BinaryRotation i num i is the Integer variable we want to modify The argument num gives the size of the shift If num gt 0 we shift to the right otherwise to the left The two functions differ in the treatment of the beginning and end of the Integer i shifting to the right positions at the beginning ar
21. Draw and move selected objects The following interactive function is a mouse interaction We can draw objects and move them The most important command for this is Sel0bj3 48 3 5 Draw and move selected objects Syntax Sel0bj drawFct messageFct numSel0bj The function drawFct contains OpenGL commands to draw the objects see chapter 7 and has no arguments It is only called on certain mouse events The calling event can be determined by Sel0bjMessage which is set automatically on calling the function drawFct Syntax Sel0bjMessage Only the events WM_LBUTTONDOWN WM_LBUTTONUP and WM_MOUSEMOVE are detected By press ing the left mouse button a certain object is selected releasing the mouse button deselects the object If an object is selected it is passed a message if the mouse is moved The function messageFct has 2 arguments of type Real where the first specifies the horizontal and the second the vertical coordinate of the mouse cursor If Sel0bj is contained in a plot2d environment the coordinates are with respect to the plot coordinate system Otherwise they denote relative screen coordinates Using the argument numSel0bj we can specify the number of objects to be generated If we want to generate several identical objects we can use the same function The objects differ by their identifier and are counted starting with 0 to numSe10bj 1 On calling the functions drawFct and messageFct the identifier is assigned to the value of
22. LoadTexture and illustrate the usage of glu Build2DMipMaps and gl TexImage2D Template function TextureLoad filename variable TextureNr gl GenTextures gl BindTexture gl TEXTURE_2D TextureNr gl TexParameter gl TEXTURE_2D gl TEXTURE_MAG_FILTER gl LINEAR gl TexParameter gl TEXTURE_2D gl TEXTURE_MIN_FILTER gl LINEAR glu Build2DMipmaps filename g1 BindTexture gl TEXTURE_2D 0 return TextureNr variable roofTile TextureLoad path filename The argument filename is of data type String and contains the full qualified path filename and filename extension of the image which is used as texture First we define a variable Tex tureNr in order to reference to the texture later on The function gl GenTextures guarantees that every texture has a uniquely determined identifier of type Integer also called ID of the texture We could also assign an identifier manually But then we have to check uniqueness by ourselves gl BindTexture binds the specified texture to the target surface Syntax gl BindTexture target texture Besides connecting texture and surface we also specify that our texture is two dimensional As soon as a texture is bound to an object every manipulation of the object takes effect on the 142 7 7 Textures texture too In this way the target acts as a representer of the texture In this introduction target is always specified by gl TEXTURE_2D The properties of the texture can be
23. Syntax initialization while condition reinitialization The while loop is a general form of a loop structure As before break and continue can be used to influence the execution of the loop body 6 5 Functions An efficient strategy to solve complex problems is to identify smaller subproblems that are easier to solve One then tries to solve the subproblems first and can try to find a solution for the overall problem from the solutions of the subproblems In this way the solutions of smaller subproblems may be used for different topics that contain the same subproblems In terms of programming languages such subproblems or subtasks are represented by func tions They act as self contained programs and can handle subproblems independently of the rest of the program They help to improve readability and reuseability of code Functions act like machines in a large factory They ask for input then work on the input in a previously determined fashion and produce a result which is returned or moved to another function In SigMath functions can be defined everywhere in the document Functions might be nested i e defined within other functions In this case the inner function is only available within the outer function see Section 6 2 2 The rules for names of variables see p 75 apply for names of functions too Functions come with their own scope variables which are defined within a function are local variables of the funct
24. accessible variables or constants All commands in the preamble are only executed once i e they are excluded from the previously mentioned infinite loop The main part contains those informations text and commands that are executed and displayed repeat 15 1 Introduction Menu point Functionality New document Opens a new empty document Open Opens a selected file Reopen Shows a list of recent files Clicking on a filename reopens the file in the editor Save Saves the document using the last specified path and file name Save As Saves the document to the specified path using a maybe different filename Close Document Closes the current document Help Displays a description of the editor Topics can be selected in the menu Select a menu item by left clicking Compile Saves and compiles the current document Opened Files Shows a list of currently opened documents in the editor By left clicking the filename the content of the file is shown Find Specify the text to find by clicking Text Searches the document starting at the current position of the cursor or starting from the beginning of the file Select by clicking Templates Inserts a selected template after the current cursor position The template can be selected by left clicking Let Open Opens the selected document in the default application spec ified by the operating system
25. and overline Syntax it bf underline text overline text Using the command without the curly brackets the modification applies to the whole text in the environment following the command With brackets only the presentation of the text in text is changed 2 4 Formatting This section deals with formatting text We already learned how to change font properties Now we want to consider alignment and spacing of text Text alignment By default text is aligned left in SigMath To alter this behavior we can set the property FlushRight to true to add right alignment to the text which results in justified text The default behavior is restored by setting the property to false 25 2 Word processing Syntax FlushRight true FlushRight false To center a certain part of the text we put it in a centerline environment Syntax begin centerline centered text end centerline A possible application of the centerline environment looks like this left aligned text begin centerline centered text on MM several lines end centerline back to left aligned text Line breaks Line breaks are performed automatically in SigMath without using hyphenation of words at the end of the line To force a line break we can use one of the commands linebreak or newline Using linebreak the text in the line is displayed justified where spacing between the words is inserted if necessary The commands and newline produ
26. argPointer 0 pointer to double The second argument is a real matrix of size 10 x 20 argType 2 DLL_ARG_DOUBLE argType 3 2 dimension argSize 0 10 argSize 1 20 argPointer 1 pointer to double The last entry of argType is always DLL_ARG_NONE All pointers in argPointer should only be used within this function On empty arrays the corresponding entry in argPointer is not a valid pointer For example if argPointer i corresponds to an empty array argPointer i i 1 This property is used by SigMath to identify empty arrays If a DLL function has more than one array as arguments then the programmer has to make sure that at most one empty array is possible The exported function or property can be used in SigMath as usual For the above function we write Identifier The prefix PROP for properties or constants allows to write 1ib Name Identifier where libName is the name of the library used in LoadLibrary SigMath actually calls the function in the DLL that returns the value of the property If a function in a DLL changes the size or layout of an array or String the DLL must contain a variable of type RECALL_RESIZE The value of the variable can be changed by the function Message_V and contains a pointer to the function that implements the method to resize the object It has the interface int RecallResize void dataP int dim int sizes Here dataP is pointer to the object which should be changed dim its new dimensi
27. be a suitable Integer value and s is the inserted text i e an expression of type String The counterpart of Insert is Delete This method allows to remove a number of subsequent characters from S starting from a specified index in S Syntax S Delete index count Here index has to be an Integer between 1 and S Size which determines the beginning of the substring to be erased The second argument count has to be a non negative Integer which gives the length of the substring If count is greater than the length of the remaining tail of S after position index only this tail is removed without warning The Replace method can be used to replace substrings in a String by another substring Syntax S Replace old new S Replace old1 new1 old2 new2 Here old is a String variable which is searched in S and is replaced by the value of the String variable new every time it occurs in S The second variant allows to replace every appearance of a substring of the form o1d1 s old2 by new1 s new2 where s is any substring Both methods return the number of replacements which have been performed The second variant is particularly useful if one e g has to exchange symbols for parenthesis efficiently The methods ToLower and ToUpper change all characters to lower or uppercase letters if pos sible All other characters remain unchanged Syntax S ToLower 87 6 Programming S ToUpper string ToLower s string ToUpp
28. can add a link to the page by writing GoToPage in the SigMath document Syntax GoToPage text label file 39 3 User interaction Here text denotes the text in the document that is displayed To emphasize its special mean ing it is displayed in a different color green by default The color can be changed by using GoToColor see Section 2 8 and Table 2 4 The second argument label contains the label of the target page If label is empty i e the link points to the first page of the current doc ument If the target is contained in another SigMath document one has to provide its filename in the last argument file This argument is optional if the links direct to a page contained in the same text document For the distribution of pages to several documents one should have the following in mind Opening a document in SigMath all code on all pages is compiled i e transformed to a displayable SigMath document even if they are at first invisible to the user The more pages a document contains the more memory it will need However one has to take care that all variables and functions needed in the page are visible that means valid see Section 6 2 2 Thus one has to decide which pages to be grouped in one document and which can or should be put into another document If one wants to use formatted text for example containing special characters in a GoToString command one has to put an inputlabel environment in front of
29. can be added text can be concatenated or we 73 6 Programming can compare boolean values Finally more complex objects can be built from elementary data types e g matrices of numbers which allow to simplify certain tasks Section 6 3 6 2 1 Constants If we speak about a constant we typically mean its concrete value e g the number 5 or the text Hello world Constants have to be treated as their actual values They can not be changed Table 6 2 shows some examples for constants Data Type Example Description Integer 98 Decimal 0x12 0X12 Hexadecimal 16 012 Octal 8 Real 1 2 2 6 11e 2 11E 2 11 x 10 1100 0 Pi 3 14159 e 2 71828 INF INF plus or minus infinity NAN Not a Number Complex I imaginary unit 5 0 2 5x 1 Bool true false true false String some text Tbl 6 2 Constants 6 2 2 Variables Variables are used to store data temporarily in memory For this purpose different storage templates exist which are called data types Those storage templates allocate memory and assign names to the memory areas for future read or write access Internally memory is organized in memory addresses which are then linked to the name of a variable An overview of all data types used in SigMath is given in Table 6 3 Initialization of variables To make a variable accessible within a SigMath documents it has to be defined SigMath not only allows but demands si
30. char xFileName int Sizes 2 155 8 Using and Writing DLLs double xD FILE fp_in 0 arguments are the filename and the matriz if GetiCharPiDoubleMatrixP arglype argPointer amp FileName amp D DLL_RESULT_OK return 1 fp_in fopen FileName r if fp_in 0 return 1 determine the size of the matriz in FileName and write sizes adapt D if Resize void amp D 2 Sizes DLL_RESULT_OK return 1 fill D return DLL_RESULT_OK Calling user defined functions If one wants to use a user defined SigMath function in a DLL we have to generate a pointer to the function This is done by using the commands GetFunction and FreeFunction Cur rently only two calls of GetFunction are possible without issuing one call of FreeFunction The actual call is performed by Message_V We will use the following definitions contained in ZS_ArgTypesDLL h typedef int RECALL_FUNCTION int int void typedef RECALL_FUNCTION RECALL_GET_FUNCTION const char typedef void RECALL_FREE_FUNCTION RECALL_FUNCTION Then Message_V may have the following implementation RECALL_GET_FUNCTION GetFunction 0 RECALL_FREE_FUNCTION FreeFunction 0 int Message_V int type void argPointer switch type E case DLL_MSG_AFTER_LOAD GetFunction RECALL_GET_FUNCTION argPointer DLL_INDEX_GET_FUNCTION FreeFunction RECALL_FREE_FUNCTION argPointer DLL_INDEX_FREE_FUNCTION break de
31. command Syntax IncludeString stringVariable The argument is a text variable that contains the source code This allows to generate code fragments on run time which may depend on user interaction or the values of other variables and to include the code at this position Example 1 1 Let us write a program that performs the rather dull job of displaying all numbers from 1 to 1000 below each other In a rather naive attempt one would actually write the 1000 lines of code Typically in programming languages one would use a loop structure more on loops in Section 6 4 2 to perform this job In this introduction we stick to pseudo code and 19 1 Introduction provide SigMath code on the right hand side Code E S From i 1 to 1000 repeat for i 0 i lt 1000 i append to S the following number S i string n and the command for a newline As a last remark the task in this little example also can be solved using the array environ ment see Section 2 9 In a similar manner as Include and IncludeString the commands Execute and ExecuteString behave Syntax Execute filename ExecuteString stringVariable Here the contents of the file filename or the variable stringVariable is assumed to be source code Its content is treated as we would have written it within a code environment see Section 1 6 2 20 2 Word processing This chapter is devoted to questions of word processing i e layouting and positioning o
32. detail in the following sections By default the class doc is activated i e we do not have to specify this class explicitely To use functions contained in other classes we have to use dot notation i e we have to provide the name of the class in front of the command separated by a dot Thus the general syntax for a command looks as follows Syntax class command In particular whenever we are using complex data types see Section 6 3 we have to use this notation If a document contains a lot of commands from one class we can preselect the corresponding class by writing use_class className Then the class className is activated i e the interpreter is parsing className instead of doc Syntax use_class NameD Class For example in parts of a document containing a lot of commands which are given by the OpenGL class we can put use_class gl in front and do not have to specify gl every time we use a command from that class This also works for dynamic classes which are imported from DLL files see Chapter 8 Apart from the classes SigMath also contains some globally defined functions and identifiers Those are e g the names of data types mathematical functions and commands to generate 71 6 Programming Class Description anim Animation array Arrays bmp Bitmaps cont Container doc Text formatting edit Text editor font Font ge Graphics expor
33. document The preamble of the example starts a thread that keeps on incrementing the value of the vari able i which is displayed The function runs as long as the Boolean variable Stop has the value false If one wants to close the document CheckThread is started This function checks the value of Stop If Stop true the function Count exits the thread terminates and the document can be closed Running user defined functions in threads may result in inconsistent behavior since they use local static variables and arguments To prevent different threads from using the same vari ables or functions at the same time we can use a so called critical section Within such a section only one thread can access certain variables at a time We define a critical section by thread InitializeCriticalSection enter it by thread EnterCriticalSection and close it by thread LeaveCriticalSection 115 6 Programming Syntax thread InitializeCriticalSection thread EnterCriticalSection identifier thread LeaveCriticalSection identifier Here thread InitializeCriticalSection returns an index of type Integer which uniquely determined the critical section The argument identifier is then the index of the previously defined critical section Example 6 10 Let us consider two functions that act on the same array variable Stop false variable A RealArray 2 20 variable CS_ID thread InitializeCriticalSection function Shuffl
34. exe and drag it to the desktop After releasing choose Create Link Double clicking the icon on the desktop will start SigMath Small SigMath examples that are used in the following sections can be found in the folder SigMath Guide which can be downloaded from the above mentioned web page 1 4 Getting started Before we show how to layout your own SigMath documents we want to make you familiar with the SigMath development tools We show how to navigate in SigMath and to use already built up sample documents After starting SigMath the user interface as depicted in Figure 1 1 is shown The top line shows the version number of the running SigMath application on the right and on the elements Deutsch and English which switch the system to German or English The bottom line contains the current date and time In the middle of the page several menu elements are listed Selecting a menu item by left clicking executes connected commands One can open a SigMath document start the editor to write and compile a new document or change general options Clicking on Close or hitting the ESC key quits the program The green words on the SigMath start page are interactive elements see Chapter 3 and are links to new pages see Section 3 1 These elements allow the user viewer to interact with SigMath They can be used to 1 direct to a new page by left clicking them 2 execute one or several commands by left clicking them 3
35. gl LINES Xgl Vertex 100 100 0 Xgl Vertex 250 250 0 gl End gl Disable gl LINE_STIPPLE 130 7 2 Primitive gl LineStipple 1 1010101010101010 gl Begin gl LINES gl Vertex 100 200 0 Xgl Vertex 250 350 0 gl End gl Disable gl LINE_STIPPLE gl Begin gl LINES gl Vertex 100 300 0 Xgl Vertex 250 450 0 gl End Fig 7 1 Lines with different line patterns In a similar way we can generate patterns to fill up polygons Syntax gl PolygonStipple pattern Here pattern is a bitmap of type gl LUMINANCE gl DEPTH_COMPONENT or gl ALPHA All three types consist of bitmaps of 128 pixels and every pixel is given by one byte 8 bit Thus we have 1288 bits to generate a 3232 pixel pattern As for the lines the pattern consists of zeros and ones which are interpreted in the same way as before Example 7 13 INSERT EXAMPLE FOR POLYGON WITH PATTERN 131 7 Computer graphics 7 2 8 Quadrics In the beginning we learned how to draw points lines and surfaces From this objects we can build more complex geometrical objects e g quadrics To ease the task of generating quadrics the glu library provides further functions to draw spheres cylinders or pointed disks These functions generate the geometrical object from primitives based on the values of the arguments Remark The way to scale move or rotate quadrics will be introduced in Section 7 4 Spheres The function glu Sp
36. in the window that contains the scenery Syntax gl Viewport x y width height The coordinates x and y mark the bottom left corner of the display area width and height specify its size The ratio of width and height should match the given ratio in the projection commands in order to prevent distortion The default values for all SigMath documents are given by the following command gl Viewport 0 0 W H where W and H denote the width and height if the window in pixels The resulting coordinates give the window coordinates 7 7 Textures 7 7 1 Basic principles Images which are mapped to the surface of an object are called tertures The mapping process is called texture mapping and the whole process is also called texturing Texturing can be used to model surfaces of objects by using image information This technique allows to add more detail to a scene without having to change its geometrical complexity Example 7 17 Imagine we want to display a paved plaza One attempt would be to generate every single stone in the pavement by primitives To generate a more realistic scenery we have to add structure to the surface of every stone Modeling may get arbitrarily involved making the task to generate a realistic image impractical Alternatively we can model the whole paved area as one primitive and map an image of a pavement to the surface we use textures This section will introduce the available types of textures and illustrate
37. is marked as done The first argument message contains the name of the WINDOWS message which can be one of the values contained in Table 3 2 Note that system keys are all key combinations containing the Alt key The arguments arg1 and arg2 have different meaning depending on the value of message All messages concerned with keyboard events pass the ASCII code of the corresponding key as arg1 On mouse events arg1 and arg2 contain the horizontal and vertical coordinates of the mouse cursor ASCII code also plays a role in determining whether a certain key is pressed 45 3 User interaction Identifier Description WM_KEYDOWN key pressed WM_KEYUP key released WM_SYSKEYDOWN system key pressed WM_SYSKEYUP system key released WM_LBUTTONDOWN left mouse button pressed WM_RBUTTONDOWN right mouse button pressed WM_LBUTTONUP left mouse button released WM_RBUTTONUP right mouse button released WM_MOUSEMOVE mouse movement Tbl 3 2 WINDOWS system messages This can be checked by IsKeyDown keys on the keyboard to numbers between 0 and 255 The first 32 characters are non visible control characters the numbers 0 to 9 have ASCII code 48 to 57 latin uppercase letters A to Z have code 65 to 90 and latin lowercase letters a to z correspond to 97 to 122 The assignment of the characters with higher ASCII code is system and language dependent For special control Syntax
38. of vertices results in drawing a quadragon There are zero one and two dimensional primitives We provide a list of all available types in Table 7 1 Although triangles and quadragons are polygons they represent very special cases within the two dimensional primitives The reason is that such objects are used quite often We give a more detailed description of these objects in the corresponding subsections 122 7 2 Primitive Dimension Object Type of primitive 0 points gl POINTS 1 lines gl LINES gl LINE_STRIP gl LINE_LOOP 2 triangles gl TRIANGLES gl TRIANGLE_STRIP gl TRIANGLE_FAN quadragons g1 QUADS g1 QUAD_STRIP polygons gl POLYGON Tbl 7 1 Types of primitives 7 2 3 Implementation of primitives In this section we give an insight in the way primitives are implemented in SigMath Let us have a look at the following template in SigMath s online help Template primitive Xgl PolygonMode g1 FRONT_AND_BACK gl FILL gl Begin gl TRIANGLES gl Color 1 0 0 gl Vertex 200 200 0 gl Color 0 1 0 gl Vertex 200 400 0 Ngl Color 0 0 1 gl Vertex 400 400 0 gl End With PolygonMode the mode of the polygon is specified Syntax gl PolygonMode side mode The argument side may take one of the following values gl FRONT gl BACK gl FRONT_AND_BACK This argument determines which side of the polygon is drawn and thus depends on the value of mode Mode can
39. path filename w h We note the two additional arguments in the function head They have to be of type Real but are reset within the function body Thus their actual values are not used The first command defines a Bitmap variable and bmp LoadPower2 loads the specified image to this variable Syntax bmp LoadPower2 filename w h This command generates a Bitmap where the side lengths equal the smallest possible power of 2 larger than the actual proportions of the loaded image The image is placed to the bottom left corner of the Bitmap and the remaining pixels are colored with black 0 Finally w and h are computed using the following formulas w width of image width of Bitmap h height of image height of bitmap The values of w and h therefore are contained in the half open interval 0 5 1 The reason for these assignments will be explained in Section 7 7 5 The remaining part of the function follows the corresponding lines in TextureLoad where glu Bind2DMipmaps is substituted by gl TexImage2D which is called using the Bitmap variable containing the image and not using the filename directly Procedural textures SigMath provides the data type Bitmap to store images To generate a new Bitmap variable we use the following Bitmap command Syntax variable image Bitmap type By default type has the value g1 RGB This generates a hxwx3 array h and w represent height and width of the image The third dimension is u
40. sceneries and thus requires special care Thus we start with the determination of front and backside 125 7 Computer graphics before treating the available types Front and backside of polygons Typically the front side of a polygon is given as the side its normal vector is pointing at Thus we have to determine the positive direction of the normal of a surface Basically there are two ways to determine this direction The first variant is called implicit definition of the normal vector which is important to draw the polygon As already noted the arguments of PolygonMode allow to determine which side of the polygon has to be drawn The determination of the sides of a polygon depends on the order of the vertices The front is deter mined by using the right hand rule We imitate the direction of rotation of the vertices using the fingers of our right hand The direction the thumb is pointing at then gives the positive direction of our normal vector The second variant uses explicit definition of the normal vector in the definition of the primi tive Using light sources this information is mandatory for the computation of the scenery and cannot be performed implicitly Since we do not go into the details here we just state the syntax to define the normal vector Syntax gl Begin areatype gl Normal x y z gl Vertex x y z gl End We will illustrate the meaning of front and backside in connection with PolygonMode in
41. source code On the difference of text and commands see Section 1 6 2 Every document in SigMath is displayed as an animation That means a lot of images every one static is displayed in high speed and gives the illusion of movement This is accomplished by parsing the current page in an invisible infinite loop and displaying it after every run The refreshment rate of the presentation is given in fps frames per second Its actual value depends on several factors e how fast are central processing unit and graphics card e what computations are contained in the current page e the frame rate of the monitor e what other programs are running in the background If the refreshment rate is too low we detect jumps in the presentation For the human eye a rate of 30 fps gives the impression of a rather smooth visualization Between the frames user interaction is processed Thus at a frame rate of 0 1 fps a user has to wait 10 seconds on processing a mouse click This way of interpreting the source code might entail unforeseen effects for programmers not used to animation We illustrate these effects by a small pseudo example The task is to add 5 to a user specified number and show the result on the screen A first attempt would look the following Code Pseudo program Request a user specified number x Input x T x gt 5 x 5 Display x Value0f x Only a small portion of programmers would define a new variable
42. take care of the fact that all points lie in one plane This class contains only one type of primitive gl POLYGON Consecutive vertices are connected to form one polygon Example 7 11 variable n 10 variable D 2 0 Pi n variable End 2 0 Pi D 2 gl Begin gl POLYGON for variable F 0 0 F lt End F D gl Vertex r cos F r sin F 0 gl End This example illustrates how to approximate a circle using a polygon The quality of the approximation is controlled by the number of vertices here n 10 As before only one object may be defined within one environment 7 2 6 Additional functions SigMath provides some additional functions like e g to draw circles or parts of the circle to ease the tasks of generating scenery This functions are contained in the directory ProgDir Shared OpenGL The functions can be accessed by using the Execute command as described on p 20 Table 7 2 on p 150 contains a list of all currently available SigMath documents together with the functions they contain 7 2 7 Line properties The style of lines can be adjusted by modifying line properties We start with the line width Syntax gl LineWidth width The argument width is of data type Real and is interpreted as pixels By default line width is set to 1 The admissible range for the line width can be retrieved with the help of the function 129 7 Computer graphics gl Get The same function also may
43. take the following values gl POINT gl LINE gl FILL On g1 POINT only the vertices are drawn g1 LINE adds connecting lines and the mode g1 FILL draws lines and fills the whole area We come back to the discussion of front and backside of polygons and hence to the meaning of side in Section 7 2 5 The current settings of the mode of the polygon are valid till the next PolygonMode occurs Thus one does not have to specify it for every primitive separately The primitive environment is enclosed by begin primitiveType and End The argument primitiveType may be one of the values listed in Table 7 1 123 7 Computer graphics Within this environment we specify a list of vertices Depending on the type of the primi tive one environment may be used to define more than one object of the same type In case of gl TRIANGLES 6 vertices result in two triangles Extending the list in type gl POLYGON results in adding vertices to the polygon Thus only one polygon can be specified within one environment As one can see in the template also the color of the primitive is declared All following objects are drawn using this color until the next Color command changes the color Thus every vertex might have another color On default settings different colors result in color gradients between the vertices which are computed during rendering process Interesting effects already occur in very simple situations OpenGL provides means to modify the comput
44. the object is determined by its vertices They play a fundamental role in computing the scenery the so called rendering process This computes light effects and color gradients depending on the vertices of an object Syntax gl Vertex P Vertex gl Vertex x y z Coordinates of a vertex can be given on two ways The first option uses a variable P of type GLPoint which has to be converted to an appropriate format using the Vertex command The second option uses the three coordinates x y and z with respect to SigMath s default coordinate system directly The origin 0 0 0 of the system is located in the lower left corner of the screen The x axis is pointing to the right the y axis to the top and the z axis points out of the monitor plane to the front Remark In this section we always refer to this default coordinate system For sceneries we introduce further coordinate systems in Section 7 6 7 2 2 OpenGL primitives Let us turn to the primitives in OpenGL As was already mentioned before the shape of a geo metrical object is defined by a list of vertices In order to specify how OpenGL has to interpret this list we have to specify the type of primitive we want to describe Example 7 1 We give a list of 4 vertices If we choose the object to be a line the 4 vertices generate 2 independent lines The vertices are left and right endpoints of the lines respectively If we choose the object to be a polygon instead the same list
45. the paragraph on Definition and initialization in Section 6 2 2 Otherwise modifications take effect on the following Input field 42 3 3 Input vectors and input matrices As an example suppose we only want to accept nonnegative entries Then we can write InputMin 0 For unconstrained variables we use INF for infinity and INF for negative infinity An upper unbounded variable is thus given by InputMax INF The third property specifies the precision and hence step sizes if the user moves the mouse up and down while pressing the right mouse button Example 3 2 The next program should allow the user to adjust the background color Therefore we need a variable of type Real for each color component red green and blue By declaring the variables every variable is assigned a starting value the variable is initialized In other words the variable R is assigned the Real constant 0 0 Specifying InputMin InputMax and InputDelta monitor the input which is realized by Input Code BackgroundColor txt variable R 0 0 variable G 0 0 variable B 0 0 begin document InputMax 1 InputMin 0 InputDelta 0 01 R Input R G Input G B Input B BGColor R G B end document Finally we can determine the last input field which was selected by the user by calling Fo cusedInput Syntax FocusedInput This property is useful when one has to process WINDOWS messages see Section 3 4
46. to fit into the rectangle default setting Using PROP_MAX_2 the scal 30 2 7 Inserting figures ing is only performed up to a scaling factor of 2 The first argument name of Image specifies the image to be displayed The image can be made available in different ways name gives the filename bmp or jpg of the file that contains the image name is ia text variable see the paragraph on String in Section 6 2 3 that contains the filename name is the name of an array see Section 6 3 2 name is the name of a variable of type Bitmap see Section 6 3 5 For arrays of numbers every element is interpreted as a grey scale value and thus a gray scale image of the size of the array is displayed The optional arguments left and top give the coordinates of the top left corner of the rectangle relative to screen coordinates 0 for left or bottom resp The parameters width and height specify width and height of the rectangle relative to screen size and depth sets the depth of the rectangle This variable is needed to detect foreground and background if objects overlap Here an object is closer to the observer if its depth is smaller By default depth is 0 We give some examples to illustrate the effect of the different values of ImageStretch Example 2 2 We start using no optional arguments That means the figure is displayed at that position in the text at which it appears in code The figure fills the whole rectangle measuring a fifth of sc
47. variable of type Real variable x 5 x is variable of type Integer 2 X y is initialized by x 2 here 7 and thus recognized as variable of type Integer variable y ct W variable Hello World t is a variable of type String The value of the variable y obviously depends on the value of x This expression is evaluated only once and assigned to the variable y as initial value Thus changing the value of x after wards does not lead to a change of the value of y The reason for that is simple after definition and initialization the variable y has the value 7 which does not depend on x any more The variable is thus assigned the resulting value of the expression and not the expression itself If a variable is used as an argument of a function there is one exception from the automatic type determination Sometimes we have to declare the data type explicitely see Section 6 5 The identifiers of variables have to be chosen according to the following rules 75 6 Programming 1 Names may contain all Latin characters the digits O to 9 and the underscore _ 2 The first character has to be a letter 3 Identifiers must not be identical to a reserved word of SigMath e g variable Table 6 4 contains a list of all reserved words in SigMath Reserved Word Description case part of the switch case structure default optional part of the switch case structure e Euler number e 2 71828 else part o
48. we draw a sphere with center lying in the origin The next sphere is generated using the same arguments as the first one Issuing the Translate beforehand moves the sphere such that its center is located at 200 200 200 Then we perform another translation scale by the factor 2 and rotate by 45 degrees around the x axis The center of the last sphere has coordinates 500 500 200 its radius is 200 and the poles are connected along the bisectrix of the yz plane All these operations are implemented as matrix multiplications from the left by the corre sponding transformation matrix see Section 7 4 Before an object is drawn its vertices are multiplied by the active OpenGL matrix In the example the coordinates of the third sphere are multiplied by the result of the following matrix multiplication identity x translation x translation x scaling x rotation 7 5 2 Matrix stack With gl MatrixMode we actually do not refer to one matrix but to a whole stack of matrices Its depth is constant in Example 7 15 Let us consider the general case here In stacks we can only access the topmost entry In case of the OpenGL matrix stack this is the current active OpenGL matrix We can use the following commands to modify the stack Syntax gl PushMatrix gl PopMatrix 137 7 Computer graphics With gl PushMatrix we can add a new matrix on top of the stack while gl PopMatrix deletes the topmost matrix The advantage of using the sta
49. y row col arrayInd The command ArrayRect returns in both versions the coordinates of a rectangle that encloses the specified entries in the table The first version returns a four dimensional vector where the first component contains the horizontal screen coordinate of the first entry of the first column the second component contains the vertical screen coordinate of this entry and the third and fourth component contain the corresponding coordinates of the last entry in the last column Using this coordinates we can draw a rectangle enclosing the whole array If we only want to enclose certain elements of the array we use the second version This requires as arguments the row and column indices of the top left and bottom right element Keep in mind that the first row or column has index 0 Using this coordinates we can draw a rectangle with OpenGL commands The following commands have to placed within the same array environment as the table The rectangle is then drawn using green color ArrayRect row1l col1 row2 col2 leftX leftY rightX rightY gl PolygonMode gl FRONT_AND_BACK g1 LINE gl Color 0 1 0 gl Rect leftX leftY rightX rightY Both commands ArrayColX and ArrayRowY require the argument coord consisting of the horizontal pixel coordinates of the positions where the columns start and the vertical pixel co ordinates of the position where the rows start Here coord is a vector whose size is determined by the number of columns o
50. 1 0 1 6 1 1 1 0 7 and 1 1 1 1 8 For two dimensional arrays we can interchange the meaning of columns and rows i e we transpose the matrix by Transpose Syntax A Transpose Traversing the sequential storage scheme of an array backwards we revert the array This can be done by using reverse Syntax A reverse For one dimensional arrays we can insert a new data value at a certain position This is performed by insert Syntax A insert j value The index j must be Integer and lie between 0 and A size 1 The data type of value has 98 6 3 Complex data types to match the data type of the array To append a value at the end of a vector we can use push_back method Syntax A push_back value As before value has to match the data type of the array The method pop_back allows to delete the last entry of a vector Syntax A pop_back To delete the first entry of a vector we can mirror the vector by reverse then execute pop_back to delete the first entry which has become the last of the reverted vector and then mirror back Further mathematical methods on arrays For arrays of arithmetical data types we can use further methods that implement mathematical functions on the arrays With Mean we can compute the arithmetical mean of all elements of the array and Sum computes the sum all elements Syntax A Mean NA Sum The methods returns the mean and sum of the array which can be stor
51. 4 Fvent handling 440 ds Bee eA EE Aha Be boe aes 3 5 Draw and move selected objects 2 2 o e e Plots in SigMath 4 1 Two dimensional plots 4 1 1 Generating plots 11 11 12 13 13 14 15 15 17 18 21 21 21 23 25 27 29 30 32 33 37 39 39 42 43 45 48 51 l 5l Contents 4 1 2 Color style and background o e AS RedraWing plots tasca A ROA a a ea es ATA ARES ii A A neds a a AL A AD e oe a 4 1 5 Interactive plots te ss 2446 4 ek cd o a a es AC NGG LOK See o iii de eS esl ai As del UNTO A YON o a et ica 4 2 Three dimensional plots 0 0 00 0 e 5 Exporting graphics 6 Programming Gila Elasses ia oth LNs a Ree A a eb Bea Ls 6 2 Data typ s Seura dost dodo A ae or ee kite ka ae eas 6 21 Constantes ona da e ee a ks 6 2 2 Variables soii ia Ba oe la A we Ba ek 6 2 3 Elementary datatypes 2 0 ee 6 2 4 Operators and expressions 2 2 ee a 6 3 Complex datatypes ee 6 3 SINES oi aee E es eters had tee By 4h on eaten pt e dera et ea da ote ae i O52 gt A a Pes cytes Gran Ge A dake ah cE pen oh Napa a ee 6 3 3 GEPON eshte 5 GPO gee Eek dodo ye Ge ES RE A ee ee R 6 34 cContaimer soe a ha gee eel dk ye GOR ES RETA eee ee GL 6 370 BitMaps wi gin ee eae de a ce A ad Pe a eS ee a ee 6 3 6 Text editors ien pat be eae Se oo Re a ee ee 6 4 Control structures e a eo ee a ee ae es G4 Av Brachetto hated Wes ho hae Ren ale daa ee pe
52. 4 elements We can store 3 x 4 12 Integer variables in M Its size is 12 its layout is 3x4 The array N has dimension 3 and layout 2x 4x0 but size 0 and thus cannot store any data In contrast to the data types we encountered before we do not have to initialize arrays ex plicitely To prevent inconsistencies all elements of an array are set to the null element of the corresponding data type by default For arithmetical data types the null element is 0 Integer 0 0 Real and 0 0 0 0 Complex For a StringArray it is given by the empty string and for Bool the value false 93 6 Programming Since every element of an array has a unique position in the array we can access the element by the index operator The position of the element is given as a d tuple to the index operator Thus to obtain the value of the element of M at position i1 id we write Syntax M il id Indexing of arrays always starts with 0 Thus for a vector given by V IntegerArray 3 V 0 returns the value of its first element and V 3 does not exist Calling this method returns a value in the form of a temporary variable which has no identifier in the usual sense of a variable Thus single values have to be accessed and stored to variables preferentially of the same data type in order to be displayed using the ValueOf command Two special elements of an array can also be easily accessed Syntax A front A back O The front method
53. 6 7 Debugging More complex programming tasks involve testing of the function and unfortunately trying to find errors SigMath s editor can be used as a debugging tool In this section we introduce two general types of errors and the debugging tools in SigMath Basically there are two types of errors syntax and semantic errors Examples for syntax er rors are missing parenthesis or typing errors in reserved words If we try to compile a document containing syntax errors using the built in text editor of SigMath compilation will abort show an error message and SigMath display the line that contains the error If the error is contained in an external document the document is opened and the corresponding line will be shown If the error occurs within one environment the line ending the environment is indicated Syntax errors are easy to fix most of the time On a semantic error the document compiles since the correct syntax is used but the program does not act as it was intended to do Reasons may be unwanted rounding issues due to the usage of wrong data types or logical errors like improper conditions in control structures In 117 6 Programming order to ease the search for semantic errors SigMath provides the BreakPoint function One vital tool to find semantic errors is to monitor values of variables Since SigMath is updating the document in every frame this might be a hard task Here BreakPoint can be used It interrupts the ex
54. As an example we give the matrix multiplication for a rotation around the positive z axis cos phi sin phi 0 0 x cos phi x sin phi y sin phi cos phi 0 0 y sin phi x cos phi y 0 0 1 0 zo z 0 0 0 1 1 1 This multiplication corresponds to gl Rotate phi 0 0 1 7 5 OpenGL matrices There are three different types of OpenGL matrices which can be activated using one of the following constants gl MODELVIEW positioning of the camera Xgl PROJECTION adjustment of the camera gl TEXTURE texture processing The first two types will be introduced in connection with the OpenGL viewing pipeline in Section 7 6 The matrix type g1 TEXTURE is treated in Section 7 7 This section only treats the general functionality and introduces matrix stacks 136 7 5 OpenGL matrices 7 5 1 Functionality of OpenGL matrices Matrices are used in connection with transformations Without going into the details of mod elview matrices we will use them in examples to transform objects The following example illustrates the transformation of a quadric Example 7 15 Xgl MatrixMode gl MODELVIEW gl LoadIdentity Xglu QuadricDrawStyle glu LINE glu Sphere 100 10 10 gl Translate 200 200 200 glu Sphere 100 10 10 gl Translate 300 300 0 gl Scale 2 2 2 gl Rotate 45 1 0 0 glu Sphere 100 10 10 The first command activates the modelview matrix and sets its current value to the identity matrix Then
55. ComplexArray of dimension 1 Since variables of type Complex consist of real and imaginary part we get a different result on the same argument Now C is a vector of length 2 with the elements 15 2 I and 3 4 I Missing elements are not automatically filled up Thus one has to provide a string consisting of an even number of numbers Besides blanks also line breaks serve as separators Blanks are used to indicate the next ele ment of the same row newlines indicate a jump to the next column of the array In the given example we thus obtain a 2 x 2 matrix of Integers with 1 0 0 15 1 0 1 2 1 1 0 3 1 1 1 4 If we try to fill a StringArray using the FillFromString method the blank is not considered as a separator since String variables may also contain blanks Thus only a new line serves as separator for elements Thus only 1 dimensional StringArrays can be filled with this method For two dimensional arrays i e matrices we can use SetBlock and GetBlock to write sub matrices or extract submatrices Syntax A GetBlock firstRow lastRow firstCol lastCol B A SetBlock firstRow lastRow firstCol lastCol B 96 6 3 Complex data types The first four arguments specify the index of the first row and first column and the last row and last column of the submatrix under consideration Using GetBlock the elements are assigned to the variable B using SetBlock the elements are filled up with the entries of B If we want to save of l
56. ExecuteString 20 EXP10 90 false 76 FILL 30 55 57 FILLED_RECT 55 FillFromString 95 96 finite state machine 121 FIXES_POINT 90 FloagDigits 90 FloatDigits 90 FloatFormat 90 91 FlushRight 25 FocusedInput 43 fontcharset 23 24 fontname 23 fontpoints 23 24 footnotesize 25 for 76 110 FrameNr 73 FramesPerSec 73 FreeFunction 156 front 94 function 76 113 GetAxesRanges 60 GetBgColor 33 GetBlock 96 GetColor 33 GetFunction 156 GetGoToColor 33 GetInputColor 33 GetInputColorFrame 33 GetInputColorSelBg 33 GetInputColorSelText 33 gl ALPHA 131 el BACK 123 el BindTexture 142 el CLAMP 143 el COMPILE 134 el COMPILE_AND_EXECUTE 134 el DEPTH_COMPONENT 131 el FILL 123 gl FRONT 123 el FRONT_AND_ BACK 123 el Frustum 140 el GenList 134 el GenTextures 142 el Get 130 el GL Q 148 el GL_R 148 el GL_S 148 el GL_T 148 el IsList 135 el LINE 123 el LINE_LOOP 125 el LINE_STRIP 125 gl LINEAR 143 el LINEAR MIPMAP LINEAR 143 el LINEAR_MIPMAP_NEAREST 143 el LINES 124 el LineStipple 130 el LUMINANCE 131 gl MODELVIEW 136 gl NEAREST 143 Index gl NEAREST_MIPMAP_LINEAR 143 gl NEAREST_MIPMAP_NEAREST 143 gl NewList 134 el POINT 123 gl POINTS 124 gl POLYGON 129 el PopMatrix 138 gl PROJECTION 136 gl PushMatrix 138 gl QUAD_STRIP 128 el QUADS 128 el REPEAT 143 el RGB 145 gl TexGen 149
57. Format string BINARY string IntFormat string OCTAL string IntFormat string DECIMAL string IntFormat string HEXADECIMAL Here BINARY gives the number in binary representation digits 0 and 1 OCTAL for octal repre sentation digits 0 to 7 DECIMAL for the standard decimal notation and HEXADECIMAL gives the numbers in hexadecimal notation digits 0 to 9 and A to F In binary representation PlusString does not take effect since the sign of the number is given by the first bit For GLPoint variables only the first three components are converted and the settings of Float Format apply The three components are enclosed by parenthesis Optionally one can specify a String that is written out in front of the row vector This string is given by the GLPointString property By default GLPointString GLPoint Syntax string GLPointString The conversion of truth values can be adjusted by giving the conversion of true and false directly The text substitutes are stored in the properties TrueString and FalseString Syntax string TrueString string FalseString To display special characters we can use their ANSI code in the ANSIToString function Here 256 symbols are listed and every symbol is mapped to an integer between 0 and 255 Syntax string ANSIToString code Here code is an Integer between 0 an 255 The entries 32 to 127 of the ANSI table are identical to the ASCII table The first 32 entries are command symbols
58. IP 55 linebreak 26 List 103 Load 88 97 LoadFromClipboard 89 LoadFromKey 89 Load Texture 142 M_AFTER_CHANGE 44 M_BEFORE_CHANGE 44 M_BEFORE_DRAW 44 M_IS_INPUT_ENTRY 44 M_MOUSE_DOWN 44 M_MOUSE_UP 44 Map 103 Mean 99 Message_V 154 MIXED 90 monospace 24 67 MouseDeltaX 47 MouseDeltaY 47 MouseX 47 MouseY 47 MultiMap 103 MultiSet 103 Multiset 103 n 86 NAN 76 92 newline 26 newpage 39 65 newpage 65 nextlevel 30 NONE 57 normalsize 25 OCTAL 91 OnDocClose 45 OnPageChange 45 OnVarChange 45 OnWinMessage 45 OpenGL 121 overline 25 PageLabel 41 ParentDir 18 19 Pi 76 Plot 51 55 56 plot2d 22 49 51 52 57 plot3d 22 51 PlotAxes 60 64 PlotBgColor 56 57 64 PlotBgStyle 56 64 PlotColor 55 64 PlotDepth 56 64 PlotHeight 52 64 PlotIndex 36 57 PlotLineStipple 55 64 PlotLineStipple 56 PlotLineWidth 55 64 PlotMargins 57 64 PlotNumPointsX2D 51 64 PlotNumPointsX3D 64 PlotNumPointsY3D 64 PlotPointSize 56 64 PlotPos 52 PlotReDraw 57 PlotStyle 64 PlotTick 60 PlotWidth 52 64 PlusString 89 91 POINT 55 Pointer 75 Pointer 79 PolygonMode 123 pop_back 99 105 pop_front 105 Position 86 PositionCl 86 Index postext 22 28 63 PrefixDLL 153 prevlevel 30 primitives 121 procedural textures 141 ProgDir 18 19 PROP 30 PROP_MAX_2 30 push_back 99 105 push_front 105 qquad 27 quad 27
59. K_DOWN 46 VK_ESCAPE 46 VK_Fl 46 VK_LEFT 46 VK_NEXT 46 VK_PRIOR 46 VK_RIGHT 46 VK_SHIFT 46 VK_SPACE 46 VK_UP 46 Void 75 VSpace 27 w 100 while 76 111 Width 72 WM_KEYDOWN 46 WM_KEYUP 46 WM_LBUTTONDOWN 46 49 WM_LBUTTONUP 46 49 WM_MOUSEMOVE 46 49 WM_RBUTTONDOWN 46 WM_RBUTTONUP 46 WM_SYSKEYDOWN 46 WM_SYSKEYUP 46 x 100 y 100 z 100
60. NT string FloatFormat string EXP10 string FloatFormat string MIXED If FIXES_POINT is chosen the numbers are written out i e the integer part is displayed followed by a dot and then followed by the number of decimal digits specified by FloatDigits The value EXP10 uses scientific representation of floating point numbers i e only one digit before the dot is shown followed by a number of decimal digits given by FloatDigits This is followed by e g e 02 which means that the previously number must be multiplied by 10 to the power 2 Finally MIXED mixes the two previously mentioned variants Here FloatDigits also takes the digits left to the decimal dot into account If FloatDigits is large enough to display the integer part fully the remaining number of digits is written in standard decimal notation Otherwise the number is written in scientific representation Converting Complex variables we can choose the way the imaginary unit is displayed The symbol or text can be modified using the property IString Syntax string IString An uppercase I is the default value The property RemoveOs allows to modify the behavior of filling up decimal digits by zeros If zeros should be shown we set it to true and to false otherwise Syntax string Remove0s Similar to FloatFormat for floating point numbers the property IntFormat allows to change the representation of Integers Syntax 90 6 3 Complex data types string Int
61. Section 7 2 9 Triangles A triangle is uniquely determined by three points and is given as a triple 1 2 3 Since three points in three dimensional space always lie in one plane no additional considerations are nec essary As with lines SigMath distinguishes three types of triangles gl TRIANGLES is the simplest type Three consecutive vertices are connected to form a trian gle Example 7 6 gl Begin gl TRIANGLES gl Vertex 100 100 0 gl Vertex 300 100 0 gl Vertex 200 300 0 126 7 2 Primitive Xgl Vertex 300 100 0 gl Vertex 400 300 0 gl Vertex 200 300 0 gl End In this example two separate triangles 1 2 3 and 4 5 6 are drawn The two triangles have two vertices in common which have to be specified in gl TRIANGLES for each triangle sepa rately On the other hand this allows to define multiple triangles in one environment gl TRIANGLE_STRIP As with gl LINE_STRIP consecutive triangles glue together The last two vertices of the previous triangle are the first two vertices of the following one but in reversed order Example 7 7 gl Begin gl TRIANGLE_STRIP gl Vertex 100 100 0 gl Vertex 300 100 0 gl Vertex 200 300 0 gl Vertex 400 300 0 gl End This definition results in two triangles 1 2 3 and 3 2 4 yielding the same object as in Ex ample 7 6 Every third vertex results in a further triangle Thus only several connected triangles can be define
62. Sel0bjInd Example 3 4 We want to generate 4 slightly translated but overlapping squares which can be dragged using the mouse Code variable X RealArray 0 variable Y RealArray 0 X FillFromString 0 4 0 5 0 6 0 7 Y FillFromString 0 4 0 5 0 6 0 7 X anim Width NY anim Height variable D anim Width 0 1 function Draw variable x X Sel0bjInd variable y Y Sel0bjInd gl Color 0 5 Sel0bjInd 1 4 0 0 2 gl Translate 0 0 1 gl Rect x D y D x D y D function OnMessage x y switch Sel0bjMessage 49 3 User interaction 1 case ui WM_MOUSEMOVE X Sel0bjInd x anim Width Y Sel0bjInd yxanim Height return case ui WM_LBUTTONDOWN Beep 1 return case ui WM_LBUTTONUP Beep 1 return begin document Sel0bj Draw OnMessage 4 end document First four horizontal and vertical arguments are read and converted to screen pixels function Draw the lowercase x and y of the current object given by Sel0bjInd are read and squares with center x y are drawn The next function OnMessage which has 2 arguments executes certain commands depending on the nature of the calling event On pressing or releasing a key a sound is played on mouse movement the position of the current object is changed In The original view after compilation is depicted in Figure 3 1 a and the result after mouse movement can be seen in Figure 3 1 b 50 a o
63. SigMath User Guide An Introduction by Michael Adam Georg Berschneider Daniel Karrasch Zolt n Sasv ri Szilvia Vock v1 0 November 2 2011 Contents List of Figures List of Tables 1 Introduction Li What is SigMath Cura Mapes a4 Pat A ARA ce dada 1 2 Basic principles ars ra ats A a de E A Se A O hed Installation 2 2 A A A Bole e rd at do dades LA Getting started icc li tee ee ee a Be Fk Be Be A A Lb Whe editote ie 2 i ase he gota me Be ae LP Gata he ee ge BE AG 1 67 SigMath dociments panii sane a o d aat Adee we a i ee a a a g 1 6 1 Basic Structur nesi moaiand euie a ee AA RR a E a 1 6 2 Text commands and comments a 1 6 3 Inserting documents oaoa 0000 cee ee Word processing 21 Environments asioi aoa we da a a dh a E a dE ee X2 Page layout os RA O REY A Re a ee Oe ee Ae BER 2 9 OMS oros te A a A eee Se Bae se Lee ee oe ae eS AG 2A ROvmMatuine ps unt bch Oe ht as he eid ea a des Qe ee Ae a eh eat Beet eee 2 5 Positioning of text eo aa eke BOQ a Hole dee 2 ee thas AO 2 6 Emumerations sin nei 4205 as Bee ed Se a ee 227 Inserting Heure d en Sede Be ee Be Ee DE hs 2 8 Colors and background aoaaa a 29 Wa bless a Asc ol ar a dees hd 210 Symbols sais age eb ee he ad Pde Bae Ae ee User interaction Sc Hyperlnks moral a pat Meche e ety ban eke ees 3 2 7 Inputtlelds ic gate gee Be A ele BGM eG ode a a ee tha es 3 3 Input vectors and input matrices 2 2 ee ee 3
64. The corresponding arrangement of the symbols in the ANSI table depends on the configuration of your operating system see p 46 in Section 3 4 Finally we deal with the converse direction i e we want to convert variables of type String to numbers Two commands are available depending on whether the result should be an Inte ger or Real variable ToInt and ToReal Syntax string ToInt radix s string ToInt radix s out The first variant returns the number the second writes the resulting number to the variable out 91 6 Programming and returns true if the conversion was successful In all cases s is a String variable which is converted to an Integer The String must be of the following form w s 0 x ddd where w stands for whitespace s for sign or 0 for an optional zero x for a lowercase or uppercase x and ddd for the digits All parts are optional the conversion only needs the the ordering If radix lies between 2 and 36 s is interpreted as a number given in the number format based on radix If radix is 0 default the first character of s determines the number format according to Table 6 9 First character Second character Interpretation 0 1 to 7 octal representation 0 x or X hexadecimal representation 1 to 9 decimal representation Tbl 6 9 Determination of the base for Integer conversion In particular no dots or commas are allowed in the String
65. Version we can obtain the current version number To increase the version number manually one uses SetVersion Syntax A Version A SetVersion Since plot objects in SigMath react to the change of version numbers see Section 4 1 3 one can use the second method to force SigMath to redraw the content of an array Methods to fill arrays The basic version to fill the elements of an array is given by writing for every element its own as signment using the index operator More convenient is the usage of the method FillFromString 95 6 Programming Syntax A FillFromString s Here s is a variable of type String and contains the data that will be stored in the array The storage pattern of the resulting array heavily depends on the layout of the array A Let us illus trate this point with the help of some examples variable R RealArray 0 variable C ComplexArray 0 variable I IntegerArray 0 0 variable S StringArray 0 R FillFromString 15 2 3 4 C FillFromString 15 2 3 4 I FillFromString 15 2 3 4 S FillFromString 15 2 3 4 R gives an array of real numbers of dimension 1 The size of the array does not take effect in the FillFromString method since it is automatically set depending on the input argument and overwrites all previous size definitions The blanks serve as separators i e the string has to be interpreted as a list of four constants of type Real Next we define C as a
66. _DRAW Before displaying an element of ArrayBlock or InputAr rayBlock the message function is called using the argument M_BEFORE_DRAW If the function returns a variable of type String instead of the element the value of that variable is displayed M_MOUSE_DOWN If the user left clicks an element of the InputArray Block the message function is called using the argument M_MOUSE_DOWN If the function returns true SigMath ignores the message i e the element will not be selected M_MOUSE_UP Same action as in M_MOUSE_DOWN but the message function is invoked after releasing the mouse button M_BEFORE_CHANGE before an element is changed by user input the message function is called using the argument M_BEFORE_CHANGE This allows to e g save the previous value of the element M_AFTER_CHANGE After an element has been changed by user input the message function is invoked using the argument M_AFTER_CHANGE This can be used to reset an element to its previous value saved by M_BEFORE_CHANGE 44 Tbl 3 1 Messages for input arrays 3 4 Event handling Further examples on message processing connected with matrix and vector inputs can be found in the online help 3 4 Event handling Another interactive part of SigMath is the ability to react to events automatically Events may be user induced e g the intention to close a document or change pages or to use an input field or might be connected t
67. a n eta p rho V vartheta 0 theta S varsigma Y varphi t iota o sigma w varpi k kappa Greek uppercase letters T Gamma A Delta Theta A Lambda Xi II Pi Y Sigma Y Upsilon Phi W Psi Q Omega Mathematical symbols Y forall J exists ast cong L perp sim lt leq oo infty gt geq 37 2 Word processing x propto O partial e bullet neq equiv approx amp otimes 9 oplus emptyset N cap U cup gt supset D supseteq notsubset C subset C subseteq in notin Z angle V nabla II prod J surd cdot A wedge V vee gt sum f int pm x times div neg lceil lfloor rceil rfloor Multilingual support vc vc ou ve E vE U oU es R vR vs S vs Miscellaneous symbols euro ldots N aleph S Im R Re p wp backslash tilde ft Re Ne amp amp 38 Tbl 2 5 Symbols in SigMath 3 User interaction User interaction is an essential part of SigMath and extends its functionality and especially is flexibility a lot Since programming interactive elements is comparably simp
68. a complex data type As most of the other programming languages SigMath provides methods to manipulate String variables efficiently 84 6 3 Complex data types and intuitively We introduce the available methods and give reference to more general functions that are defined in the string class Throughout we assume a variable S to be initialized like in the following example variable S The arithmetical operator is also defined for operands of type String It concatenates two objects of that type to a new object of type String The compound assignment appends the right operand to the end of the left operand while the right operand remains unchanged compare Table 6 6 in Section 6 2 4 variable S1 Hello S1 World After execution of those two lines of code the variable S1 has the value Hello World The reserved memory for a String variable has to be adjusted dynamically after initialization S1 contains 5 characters and after concatenation and assignment its length has increased to 11 characters occupying more memory than before We interpret objects of type String as sequences of characters Every character even a blank has a uniquely determined position its index and can be accessed by the index operator O SCi returns the i th character of S Indexing starts with 1 and only positive integers are allowed as arguments The length of a String is given by the property Size It is a read write property i
69. able is not used outside of the loop Together with the reinitialization expression the condition leads to four repetitions of the loop body Reinitialization may also contain more complex updating expressions for variable i 2 i lt 10 i i 7 2 Sum i This time the loop body is only executed twice since i has the value 16 before entering the loop a third time As with if else structures we can omit the enclosing brackets if the loop body consists of one line only 110 6 4 Control structures The loop header The initialization part of the loop header may have two forms Either we define a new counting variable in the loop header which then has restricted scope or we use a previously defined variable In the second case the counting variable remains visible after the loop and its value is the resulting value of the last reinitialization Within the loop body we have unlimited read and write access to the counting variable While read access does not influence the behavior of the loop write access on the counting variable may result in uncontrollable behavior One might even get stuck in an infinite loop Note that reinitialization must not be closed by a semicolon SigMath does not support the usage of more than one counting variable within one for loop and the condition as well as the reinitialization must not consist of more than one command Aborting the loop By default execution of the loop terminates as soon as the cond
70. ally useful for counting variables and its usage may be preferred to i i 1 or i 1 The last variant has the same effect though Comparisons Comparison operators allow to compare the values of two expressions The result of such an comparison is of type Bool All available comparison operators are listed in Table 6 6 together with an example Comparisons are used in conditions and loops and may be used to decide between two or more alternatives If one tries to compare mixed expressions both expressions are automatically converted to the more complex data type then compared within this class in a mathematically sound way The values 2 and 2 0 for example have different realizations and are not equal with respect to the system After conversion SigMath recognizes them as the same number On arrays different rules apply see Section 6 3 2 for more information Logical operators Logical operators allow to combine different Boolean values The operands of a logical operator thus have to be of type Bool Thus also expressions which return a value of type Bool are 82 6 2 Data types allowed There are three logical operators The two binary operators AND amp amp and OR combine two Boolean values the unary operator NOT negates the value Mind that OR is not exclusive i e as soon as one of the two operands has the value true the value of the is true We state all possible connections of Boolean values in Table
71. an do that we have to enable texturing Syntax gl Enable gl TEXTURE_2D Now texture coordinates can be specified before the corresponding vertex in the primitive definition Syntax gl TexCoord s t gl Vertex x y Here texture coordinates are in normalized form i e 0 0 denotes the top left corner of the texture and 1 1 the bottom right corner We close the texture mapping process by deac tivating texturing Syntax gl Disable gl TEXTURE_2D Example 7 19 EXAMPLE MAP TEXTURE TO SQUARE We already learned that the wrap parameters see p 143 control whether texture can be repeated The number of repetitions is determined using the relative texture coordinates If we assign s to x the texture is repeated s times until we reach x The same applies to the vertical direction Example 7 20 MAP TEXTURE REPEATEDLY TO SQUARE 147 7 Computer graphics A texture coordinate smaller than 1 results in displaying only a part of the texture If the texture coordinate is larger than 1 we repeat the texture Remark This should explain why we set the values w and h in the function TextureLoad in the prescribed way see p 145 Texturing quadrics Since we do not define the vertices of a quadric explicitely we cannot use the method described in the previous paragraph Instead of defining the mapping pointwise we use a function to generate texture coordinates for the texturing of quadrics Syntax glu QuadricTextur
72. and Example 3 3 3 3 Input vectors and input matrices If we want to display and modify array variables we have to use InputArrayBlock instead of Input For the moment an array variable is just a way to store matrices or vectors in SigMath A precise definition is given in Section 6 3 2 With InputArrayBlock we can allow or forbid to change entries of the matrix 43 3 User interaction Syntax XInputArrayBlock arrayName XInputArrayBlock arrayName mHandler The argument arrayName provides the name of a variable of type array of dimension 1 vec tor or 2 matrix in analogy to Name0f Variable in the Input command The size of the vector or matrix is not specified beforehand and can depend on user input The array environment which contains the InputArrayBlock command is refreshed in every frame In the second ver sion mHandler gives the name of a function that processes certain messages The function must be of the following type function mHandler message i j 1 Here the Integer variables i and j give row and column indices of the current element and the variable message may take one of the values listed in Table 3 1 For vectors i 0 Value Description M_IS_INPUT_ENTRY Before displaying an element of InputArrayBlock the mes sage function given by mHandler is invoked with argument M_IS_INPUT_ENTRY If mHandler returns false the value of the entry can not be modified by the user M_BEFORE
73. ariable besides its data type is its scope and visibility area The scope of a variable is given by all environments which allow read and write access to the variable The visibility area is concerned with names of variables that are used several times Since scope and visibility also apply to functions all statements in this section also apply to functions With regard to scope of a variable there are two options either the scope of a variable is the whole document or an environment In the first case the variable is called global in the second case its local Variables that are defined within an environment are no longer accessible as soon as the defining environment is left All functions and environments we encountered in section 2 1 limit the scope of a variable Local variables of an environment are valid in all sub environments The only exception to this rule connected with nested environments is the following every time a global and a local 77 6 Programming variable of the same name are specified the local variable hides the global one The global vari able remains valid but is no longer visible since it is hidden by the local one This guarantees uniqueness of variable names at each time point of code execution since this rule prohibits the visibility of two variables with the same name In order to prevent the described effect we suggest to define variables as local as possible Thus variables of the same name may
74. ath in the following way Include DocDir Examples IncludeFiles Include txt The commands DocDir DocName ProgDir and HelpDir are used as in the examples For navigation in upper level directories we use the command syst ParentDir 18 1 6 SigMath documents Syntax syst ParentDir level dirName ParentDir requires two arguments the directory dirName we start with and the number of levels level we want to move up in the directory tree The argument level has to be a non negative integer The level 0 returns dirName itself We will illustrate the usage of Include commands when we learn more on page layout templates in Section 2 2 Command Directory DocDir Directory that contains the current document DocName full qualified path name directory and filename of the current document ProgDir Directory that contains the executable SigMath exe HelpDir the help directory depends on the chosen language version ParentDir parent directories Tbl 1 3 Commands that return file locations A second variant of the Include command accepts as an argument the name of a text block To do this one has to define a text block and fill it Syntax begin textBlockName Text and commands end Later the content of this text block can be inserted using its key textBlockName Include textBlockName at this point Another rather unusual way of including code is given by the IncludeString
75. ation of color gradients We will not treat further methods and restrict our considerations to the default behavior only 7 2 4 Points and lines gl POINTS are the only representers of zero dimensional primitives On contrast to vertices points have a diameter of one pixel Example 7 2 gl Begin gl POINTS gl Vertex 100 100 0 gl Vertex 300 100 0 gl Vertex 150 300 0 gl End One vertex always generates one point Thus within one environment arbitrary many points may be defined One dimensional primitives are lines A line is determined by its left and right endpoint We write them as pair left right and identify the points with their position in the list of vertices For example 1 2 defines a line that connects the first vertex to the second OpenGL distin guishes three types of primitives for lines gl LINES is the simplest version any two consecutive vertices are connected as left and right end point of a line respectively Example 7 3 gl Begin gl LINES gl Vertex 100 100 0 gl Vertex 300 100 0 gl Vertex 300 100 0 gl Vertex 200 300 0 gl Vertex 200 300 0 gl Vertex 100 100 0 124 7 2 Primitive gl End Here three independent lines are given Those are 1 2 3 4 and 5 6 The lines form a triangle One environment may contain arbitrary many lines gl LINE_STRIP On determining n vertices the second up to n 1 th vertices are interpreted as left an
76. be used to detect the smallest difference that is realizable the so called granularity Both values ranges depend on the installed OpenGL implementation on the system Syntax Xgl Get LINE_WIDTH_RANGE Xgl Get LINE_WIDTH_GRANULARITY The first request on the range returns a one dimensional RealArray with two elements the first field gives the minimal and the second field the maximal line width The granularity is also given as a Real constant Line smoothing can be activated or deactivated using the following commands Syntax gl Enable gl LINE_SMOOTH gl Disable gl LINE_SMOOTH If line smoothing is activated the line width is automatically rounded up to the next integer Using gl LineStipple allows to define line patterns Syntax gl LineStipple factor pattern The argument factor is of type Integer which determines the way the pattern is stretched The argument pattern requires a String of length 16 which consists of 0 and 1 represent ing empty and drawn pixels respectively The usage of line patterns has to be activated or deactivated using the following two commands Syntax gl Enable gl LINE_STIPPLE gl Disable gl LINE_STIPPLE Example 7 12 The following lines of code generate three lines The first line is dashed the second appears as dotted line and the third is a solid line See Figure 7 1 gl LineStipple 1 1111000011110000 gl Enable gl LINE_STIPPLE gl Begin
77. case available It is always executed if none of the alternatives is satisfied The default case should always be at the last position If we close all case blocks by break the default case is executed if no other case is executed The syntax for a switch case structure reads as follows Syntax switch expression case Al commands break case A2 commands break default commands break Example 6 4 In this example we want to check if x 5 equals 3 10 or none of the two cases We store the result of this comparison in a text variable in words switch x 5 case 3 Display The result is 3 case 10 Display The result is 10 break default Display The result is not 10 break First the simple expression x 5 is evaluated If the result is 3 the first case applies If x 5 10 is satisfied the second case applies and we terminate the switch block by a break In this example the first case is missing the break command If the first case applies the default case is evaluated too In the case of x 5 10 we do not reach default Another interesting effect is illustrated by the following example 108 6 4 Control structures switch x case 1 x case 2 x case 3 x What is the value of x after executing the switch case structure if the initial value is 1 The correct answer is 4 since by incrementing x all cases apply and no break command is used Multiple bran
78. ce lines with normal spacing and left aligned text The last two commands are in fact identical Syntax Linebreak newline Since SigMath s display is not restricted to a limited sheet of paper but on a basically infinite paper roll by scrolling the screen there is no page break in the usual sense Instead we can add links to new pages Links will be treated in Section 3 1 in Chapter 3 Spacing If we want to modify the line spacing we set the property BaseLineStretch to a factor that shrinks the line space lt 1 or enlargens it gt 1 26 2 5 Positioning of text Syntax BaseLineStretch For example using BaseLineStretch 2 we double the line spacing Furthermore the commands VSpace and HSpace insert vertical and horizontal space respec tively As arguments we specify the spacing in terms of screen height or width Syntax VSpace vertDistance HSpace horDistance The arguments vertDistance and horDistance are of type Real Using word1 HSpace 0 05 word2 we generate a horizontal spacing of 5 of screen width between the two words Analogously we can add vertical spacing between e g two paragraphs by using VSpace Often used horziontal spacings can be inserted by special commands which are contained in Table 2 3 The factors are given are with respect to the height of the current font providing comparable behaviour for all font sizes The negative factor 3 18 corresponds to a shortening of
79. change the value of a variable hold the right mouse button and move up or down OR left click the element then enter a new value in the appearing input frame 13 1 Introduction sigmaeno o7a D PAS MOTZ ONO LS 43 100 Fig 1 1 Starting up SigMath remove previous value by using the DEL key 4 move objects press left mouse button and move the mouse Navigation within documents is similar to navigation on web pages By clicking input fields one can change pages Navigation via keyboard involves the following keys e ESC closes the current page if the documents allows it e Using the cursor keys or mouse wheel one can move text that is not fixed to a certain position e If the page contains hidden elements e g for a presentation they can be shown by PageUp and PageDown If an animation is running on the screen it can be stopped by pressing CTRL Space simultaneously if the documents allows it For example the time on the start page which is an animation can be stopped in this way Hitting CTRL Space again or left clicking in the window the animation is resumed 1 5 The editor Since SigMath documents are platform independent text files basically every text editor can be used to write SigMath source code Nevertheless it might be worth considering to use the built in text editor By a simple mouse click a SigMath document can be compiled and run directly On the SigMath start page Figure 1 1 the editor op
80. ches can be nested as simple branches SigMath does not restrict the type of expressions used in the switch condition The compar isons in the case statements are performed using automatic Al type conversions as indicated on p 81 Thus 5 5 0 is true If no type conversion is possible the result is automatically set to false Thus comparisons like 5 Text are syntactically correct This feature turns switch case structures into a very flexible programming tool Example 6 5 We already encountered the switch case structure in connection with func tion calls by interactive elements see Example 3 3 A function f may be called by different DoString commands on different pages of a document see Section 3 1 Depending on the calling element we can execute different code blocks in the function The last example illus trates the usage of switch case structures in functions In general the function f has e g the following structure function f switch SelectedDo case identi commands case ident2 other commands default commands 6 4 2 Loops Loops can be used to execute certain code blocks several times If the number of repetitions is known before we use counting loops whereas while loops allow a little bit more flexibility 109 6 Programming A loop structure consists of two parts loop header includes the type of loop the number of iterations or conditions on the number of repetitions whereas the loop bod
81. ck structure is that we do not have to reset the transformation matrix to identity for a new transformation If we need the cur rent active matrix later on we just push a new matrix to the stack This allows to perform a transformation which is independent of the former ones without having to discard the previous information Example 7 16 Xglu QuadricDrawStyle glu LINE Xgl MatrixMode gl MODELVIEW gl LoadIdentity Xgl Translate 200 200 200 glu Sphere 100 10 10 gl PushMatrix gl LoadIdentity gl Translate 600 600 0 glu Sphere 100 10 10 gl PopMatrix gl Scale 2 2 2 glu Sphere 100 10 10 In this example the second sphere has its own transformation matrix without having to delete the first one The latter is turned active by deleting the additional matrix from the stack on gl PopMatrix This results in the fact that the centers of the first and last sphere coincide 7 6 Scenery This section is devoted to the generation of scenery the so called OpenGL Viewing Pipeline Composing a scenery is performed in several steps that will be explained in detail First we give a schematic view on the whole process First the object coordinates are multiplied by the modelview matrix g1 MODELVIEW Object coordinates are the coordinates which were specified in the definition of the primitives used for this object They are referring to the usual global coordinate system having its origin in the bott
82. cts lying outside this cuboid are neglected for the generation of the scenery Besides the clip planes we have to specify the type of projection There are two variants available 139 7 Computer graphics Parallel projection the camera focuses on an infinitely distant point That means the dis tance of an object to the observer does not affect the size of the object Distance is only used to detect if one object is blocked from sight by another object in the front This projection results in a two dimensional view on the scene Syntax gl Ortho left right bottom top near far The arguments are used to define the six clip planes Starting from the previously fixed camera position we specify a minimal and maximal distance to the camera This generates a rectangle having the camera in its center Finally near and far specify near and far distance which turns the rectangle into the cuboid The orientation of the cuboid in space is given by the direction of view For the distances the following rules of sign apply distances to the right to the top or to the front in direction of view must be positive distances to the left to the bottom and to the back must be negative The arguments near and far may take positive or negative values of type Real INSERT EXAMPLE OF ORTHOGONAL PROJECTION The default values for all SigMath documents are given by Xgl Ortho 0 W 1 0 H 1 1000 1000 where W and H denote the window width and wi
83. d by the priority of the operators A second options is to enclose parts of the expressions in functions In the following we study the effect and some specialties of the operators listed in Table 6 6 Sign The sign operator is an unary operator and should not be confused with the addition or sub traction The operators and add a sign to variables of arithmetical data type A positive sign does not have to be written explicitely Arithmetical operators The arithmetical operators and are all binary operators Their operands may be any variables of arithmetical data type If all operands belong to the same data type the effect of the arithmetical operators is intu itively clear Addition subtractions multiplication division and exponentiation are executed within the data type Division of two integers produces the integer part of the result if the result would be rational Thus e g 5 2 2 and 5 0 2 2 5 80 6 2 Data types Priority Operator Description Example 1 increment decrement A 1 sign unary 5 1 bitwise complement 15 1 logical negation IB 2 i exponentiation 5 0 7 2 0 3 multiplication division 5 2 4 o oS addition subtraction 5B 2 4 concatenation abc str 5 lt lt smaller smaller or equal i lt 5 gt gt greater greater or equal i gt 5 6 l equality inequality i 7 amp amp bitwis
84. d in one environment gl TRIANGLE_FAN The first vertex is common vertex to all triangles in the primitive It is always the first vertex in any triangle The second vertex in every triangle is given as the last vertex of the previous triangle The third vertex of one triangle is then the following vertex i e for the second triangle the fourth vertex in the list and for the third triangle the fifth in the list and so on Only the first triangle is given by the first three vertices in the list in the mentioned order This type may be used to draw a fan therefore the name or to draw a wheel with spokes Example 7 8 gl Begin g TRIANGLE_FAN gl Vertex 200 300 0 gl Vertex 100 100 0 gl Vertex 300 100 0 gl Vertex 400 300 0 gl Vertex 300 500 0 gl Vertex 100 500 0 gl Vertex 0 300 0 gl Vertex 100 100 0 gl End 127 7 Computer graphics This example results in a wheel consisting of 6 triangles 1 2 3 1 3 4 1 4 5 1 5 6 1 6 7 and 1 7 8 As before only one object can be defined in one environment Quadragons Quads A quadragon is defined via its 4 vertices Now 4 points in three dimensional space do not have to lie in the same plane but in a quadragon they have to Violating this condition may result in incorrect display of the quadragon Since the condition is not checked automatically we have to take care that our vertices meet the requirement ourselves There a
85. d right endpoints respectively This results in one connected line consisting of n 1 pieces The first and last vertex give starting and endpoint of the polyline Example 7 4 gl Begin gl LINE_STRIP gl Vertex 100 100 0 gl Vertex 300 100 0 gl Vertex 200 300 0 gl Vertex 100 100 0 gl End Those 4 vertices result in the lines 1 2 2 3 and 3 4 The result matches the object in Example 7 3 but requires only 4 vertices One environment may contain only one polyline gl LINE_LOOP In contrast to the polyline gl LINE_STRIP the first and last endpoint are also connected by a straight line which results in a closed polyline of n vertices connected by n lines Example 7 5 gl Begin gl LINE_LOOP gl Vertex 100 100 0 gl Vertex 300 100 0 gl Vertex 200 300 0 gl End This generates the same result as in Example 7 5 saving another vertex Now the lines 1 2 2 3 and 3 1 generate the triangle The last vertex is the left endpoint of the last line and the first vertex its right endpoint Thus every vertex is left and right endpoint of a line Multiple definitions in one environment are not allowed 7 2 5 Surfaces Surfaces are two dimensional objects and thus the most complex primitives in OpenGL Tri angles and quadragons are the most frequent polygons and thus have a special role under all polygons Every surface has a front and a backside This distinction has effect on computing
86. d to our document Include ProgDir Shared GUI StdPageStyle txt begin document begin HeadLeft text on the left end begin HeadRight text on the right end Include ProgDir Shared GUI StdHeadFoot txt Std_BG end document We already learned the meaning of the Include commands in the first and fifth line The inserted files contain page layout commands Exchanging text on the left and text on the right by new text it will be written in the top line on the left and on the right hand side respectively Finally the command Std_Bg generates the white frame and color layout of the page Other page layout templates available in the templates menu modify the look of the frame and add the possibility insert footnotes The last template in the list simple with preamble just defines a page with default background color and without frame This template does not include further SigMath documents but writes all necessary commands to alter background or text color and positioning of the text directly to the preamble of the document instead In order to customize the page format we have to change the values of the three properties LeftMargin TopMargin and TextWidth Syntax LeftMargin TopMargin TextWidth All three commands can be placed anywhere in the code but TextWidth is only treated during compilation The properties take values between O and 1 and give distances proportional to screen wid
87. de GQ ieee wa A 6 4 2 OOPS A he are bre aM Ra a 6 07 Functions 62 lt aids A eee A Be tech ee WS ES 6 6 Multithreading 0 0 22 State ay poe a Bo p eh ee Nd A ee aS 627 Debugeing ti A Eee eed Bo eh be ele Oo A ee 7 Computer graphics TA Whatis OpenGL Voir oe ee a ee ee ee 2 PLE si of es Vaal hae en aon cy Seating Sky See Se Ee ee ee RDA NOTICES ad it cay Patio he E ER ee Bee OE ee ee 1 22 OpenGli primitiveses tn ea Re Se eo ee Fa Ate eo al 7 2 3 Implementation of primitives 2 200000 00 4 TA Points and Mines ote e a a Be Fee Bet Ih nS He 2d A ea ee Be ghd Be Bich Be Bae a id aa 7 2 6 Additional functions 02 000 ee eee ee T 2if Lime properties srs sdaide g eo Re a a a a a 12 87 QUAdriCS 2 843 A ke BA eA Ee ek ae he ed 7 2 9 An extended example 65 71 71 73 74 74 78 80 84 84 92 99 102 105 105 105 106 109 112 114 117 Ta Working with ists vw 2 6 grag ios d aipua a 7 3 1 Generating lists 13 2 Callingilists iia a 7 4 Transformations e CAM Sealing ada Pg ae Bb Bode Bee Se Oe YG 4 27 Translation seo dd aa a Go ea TASS E O fo as ASR 7 5 OpenGL matrices 2 2 0 0 00 eee ee ee 7 5 1 Functionality of OpenGL matrices 7 5 2 Matrix stack m cosi yam ers aa ee aa es A Y ssa hog Ge DS As ea OE ee AE ak 7 6 1 Positioning of the camera 0 27 Projections 4 4 2 4 8 ea need is ls bes 6 3 Viewport a p
88. e while Stop if mod syst ProgTime 2 lt 1 A Resize 2 40 else A Resize 2 1 thread BeginThread Shuffle begin document thread EnterCriticalSection CS_ID beginfarray ArrayBlock A end array thread LeaveCriticalSection CS_ID end document The critical section takes care that the function Shuffle changes the layout of the array A while ArrayBlock displays the value of the array Example 6 11 As a last example we want to run one function SimpleTask several times and simultaneously To do this we have to create several copies of the function and maybe of certain variables in a template 116 6 7 Debugging variable results i 0 function SimpleTask iQ Here i acts as wildcard character that will be replaced by a number later on We store this template to a file called TemplateSimpleTaks To create several copies in our document we add the following lines variable Template Template Load TemplateSimpleTask variable RegID CurrentRegID for variable i 1 i lt NumberCopies i variable string Temp Temp Template Temp Replace i string ToString i ExecuteString RegID Temp In this way we created NumberCopies versions of the function SimpleTask and of the variable results Now we can run each copy in a different thread for variable i 1 i lt NumberCopies i thread BeginThread SimpleTask string ToString i
89. e we can also assign a new value to this property Syntax S Size If the new length is shorter than the actual length of the String the pending characters are cut off If the length is larger the text is filled up with blanks If we assign a negative value the length of the String is automatically set to 0 Since we cannot simply add reserved symbols in SigMath to strings the string class contains substitutes for such symbols If we want to write a filename with its path to a variable we would like to write e g S Images File bmp This would return an error message since is a reserved symbol The backslash has to be generated by Backslash and quotation marks are inserted by Quote instead Syntax string Backslash string Quote Thus the correct format of the above command is 85 6 Programming S Images string Backslash File bmp Text formatting commands like line breaks or tabulators can be inserted by n and t Syntax string n string t Both commands increase the size of a string by 1 Sometimes it is necessary to detect whether a given string S only contains numbers or letters For this the following three methods are available Syntax S IsAlNum S IsAlpha S IsDigit Here IsAlNum return true if S only contains alphanumeric characters i e numbers or letters IsAlpha checks for letters only and IsDigit checks for numbers Methods to modify strings To extract a substring f
90. e AND 5 amp amp 2 8 bitwise OR 5 2 9 28 logical AND B1 amp amp B2 10 logical OR Bi B2 11 assignment i 5 11 compound assignments x 2 t SS x 2 Tbl 6 6 Operators in SigMath For mixed expressions i e where the two operands belong to different data types the basic behavior is as follows the expression returns a variable of the more complex of the data types of the two operands Thus automatic type conversion is performed resulting in the listed return data types in Table 6 7 operator Integer Real Complex Integer Integer Real Complex Real Real Real Complex Complex Complex Complex Complex Tbl 6 7 Automatic type conversion for arithmetic operators Another specialty is concerned with assignments If the value of an expression is assigned to an already existing variable two data type conversions may occur a first conversion towards a more complex data type may be needed due to the arithmetical operator a second conversion might be necessary to store the result in the data type of the variable which might be less complex than the resulting type Thus for example if an arithmetical operator yields a result of type Real and the target variable is of type Integer the decimal places of the Real variable are cut off and only the resulting integer part is assigned to the Integer variable 81 6 Programming Assignments In a simple assignment the operator
91. e condition This glu function enables or disables the generation of texture coordinates for quadrics de pending on the value of the Boolean argument condition If it is disabled all quadrics are automatically drawn using the current texture bound to it The number of repetitions can not be controlled with the help of the texture coordinates s and t directly By default glu QuadricTexture generates the texture coordinates in such a way that the texture is mapped only once to the object Since texture coordinates are multiplied by the texture matrix before displaying them we can use this matrix to modify the coordinates Example 7 21 INSERT SPHERE WITH TILED TEXTURE On scaling all s values of the texture coordinates are multiplied by 4 and all t values by 3 The tiling of the square in Example 7 21 may also be generated by using the texture matrix Reflection mapping Reflection mapping allows to generate interesting mirror effects in a simple way The effect can be interpreted in the following way the texture is mapped to the inside of a sphere that encloses the object Then the texture is mapped towards the center of the sphere This is done using spherical coordinates but we do not give the details here On loading a texture we can specify a function that generates the texture coordinates Syntax gl TexGen texCoordinate genFunction genParameter texCoordinate determines which texture coordinate is modified The following va
92. e horizontalScaling equals the number of slices the radialScaling gives the number of rings the disk consists of The function glu PartialDisk is a modification of this command Syntax Xglu PartialDisk innerRadius outerRadius horizontalScaling radialScaling initialAngle cutSize This function allows to draw segments of a disk We need two additional arguments in comparison to glu Disk The initial angle of the segment and its size Both are given in degrees and measured in clockwise direction One can imagine that we cut a slice out of the pie Properties of quadrics With the help of the command glu QuadricDrawStyle we can specify certain general properties to draw quadrics Syntax Xglu QuadricDrawStyle property The possible values of property and its meaning are the following glu FILL surfaces are filled glu LINE boundaries of surfaces are drawn glu POINT only points are drawn glu SILHOUETTE only edges are drawn Since quadrics are built up from primitives all specifications in gl PolygonMode take effect on the way they are displayed see Section 7 2 3 The following applies to quadrics if both properties are set the one that draws less is applied Example gl PolygonMode gl FRONT_AND_BACK gl LINE glu QuadricDrawStyle glu FILL In this example only lines are drawn We already mentioned the normal vector of a surface in connection with front and backside of polygons Every quadric consist of surfaces and
93. e result of a comparison is false The assignment operator is defined for container variables too The data types of the operands have to match i e no automatic type conversion is performed Afterwards both containers equal the right operand Using swap we can exchange the content of two containers of the same type After issuing A swap B the variable A contains the contents of B and vice versa 6 3 5 Bitmaps 6 3 6 Text editors 6 4 Control structures By default all commands in a SigMath document are executed in the order they are written in the document This section introduces control structures that allow to break out of this sequential execution We distinguish two structures branches or loops Using a branch allows to execute different commands depending on the value of a condition Thus there are different ways to traverse the document A loop can be used to repeat certain lines of code The number of repetitions depends on a condition which has to be specified by the programmer 105 6 Programming 6 4 1 Branches In SigMath we distinguish between simple branches which are realized by if else structures and multiple branches which are realized by switch case structures We start with simple branches The simple branch Simple branches allow to execute different code depending on the value of a Bool expression called condition The branch is realized by using the if command followed by a condition If the condition evalua
94. e vacant and the corresponding number of bits on the right are cut Shifting only fills up the 83 6 Programming vacant positions by 0 and drops the bits at the end Rotation copies the trailing bits from the end to the vacant part at the beginning Code Binary representation Decimal representation i 0000000000000000000000000000 10 01 9 BinaryShift i 2 160 0000000000000000000000000000 10 2 BinaryRotation i 2 01 0000000000000000000000000000 10 231 2 The framed parts of i can be found at the specified positions While 01 is lost on the shift it is placed at the beginning of the variable on rotation 00 shows the filling with zeros The effect of the commands is not in situ i e the value of i remains the same the return value is placed in a different part of the memory and can be stored to a new variable by writing j BinaryShift i 2 There is no distinction between sign mantissa or exponent Thus the sign of an Integer may be changed by bit manipulation No type conversion is performed i e all operands have to be of type Integer 6 3 Complex data types A complex data type is built from elementary data types and is internally realized as class in SigMath see Section 6 1 In this case classes act as templates They contain properties and methods i e functions that allow to manipulate the properties A variable of complex data type is an instance of the class that has all prop
95. ecution of the program and returns the values of the active variables in the current environment On a mouse click the execution is resumed until the next BreakPoint is reached After the last BreakPoint in a document the execution is repeated up to the first BreakPoint in the document Syntax BreakPoint level name The arguments level and name are optional Using level we can determine the number of higher level environments which will be monitored By default the argument is 0 i e only the variables of the active environment are displayed variable X 0 codef for variable i 1 i lt 2 i BreakPoint 2 In this example level 0 only contains the counting variable i Level 1 applies to the code block that does not contain further variables Specifying level 2 also displays the value of the vari able X If one uses several BreakPoint commands it proves to be useful to assign names to them This makes it easier to identify the corresponding positions in the source code Example 6 12 Let us implement the Newton algorithm to determine a root of a given polynomial in SigMath It is an iterative procedure with the following updating rule jee ifn 0 Tic Ln 1 se ifneN While the iteration is running we want to monitor the values of x and f 1 The polynomial is given by fa 6x da 24 and we choose x 3 5 as initial value For this we define a global variable x that contains one root of t
96. ed to another variable 6 3 3 GLPoint The data type GLPoint is comparable to a one dimensional RealArray with four elements As indicated by the name this data type may be used to store information on points in three dimensional space This data type is especially useful in connection with OpenGL commands and simplifies syntax In this section we focus on the general description and usage of this data type Its connection with OpenGL programming is postponed to Chapter 7 Definition The following example shows the correct way to define a variable of type GLPoint The argu ments are interpreted as x y z and w coordinates of a point For the interpretation of the last coordinate we refer to the chapter 7 on OpenGL Syntax variable name GLPoint x y z w variable name GLPoint x y z variable name GLPoint x y 99 6 Programming Xvariable name GLPoint x variable name GLPoint Missing values in the first three coordinates are automatically set to 0 0 a missing fourth com ponent w is set to 1 Using the dot notation we can access every single coordinate of a GLPoint variable Syntax P x P y P z P w Here we assume P to be a variable of type GLPoint Then P x returns the first P y the second P z the third and P w the last coordinate of P The command Input Value0f allow to ways to access its values We can either specify a single coordinate or the whole point The following example il
97. el TexImage2D 142 gl TexParameter 143 el TEXTURE 136 el TEXTURE_2D 143 el TEXTURE_MAG_FILTER 143 el TEXTURE_MIN_FILTER 143 gl TEXTURE_WRAP_S 143 el TEXTURE_WRAP_T 143 gl TRIANGLE_FAN 127 gl TRIANGLE_STRIP 127 gl TRIANGLES 126 gl2ps FontName 67 GL_TEXTURE_GEN_MODE 148 GLPoint 75 99 glpoint 84 113 GLPointString 91 glu 121 glu Build2DMipMaps 142 glu Cylinder 132 glu Disk 132 glu FILL 133 glu LINE 133 glu LookAt 139 glu OUTSIDE 134 glu PartialDisk 133 glu Perspective 140 glu POINT 133 glu QuadricDrawStyle 133 glu QuadricTexture 148 163 Index glu SILHOUETTE 133 glu Sphere 132 GoToColor 33 40 GoToPage 39 151 greek 24 Height 72 HelpDir 18 19 HEXADECIMAL 91 hline 33 HSpace 27 Huge 25 huge 25 I 76 78 if 76 78 106 Tgnorelf 107 Image 30 32 57 ImageStretch 30 32 Include 18 21 65 IncludeString 19 20 INF 43 76 Input 42 43 100 InputArrayBlock 35 43 44 97 InputColor 33 41 InputColorFrame 33 InputColorSelBg 33 InputColorSel Text 33 InputDelta 42 inputlabel 22 40 InputMax 42 InputMin 42 Insert 87 insert 98 105 Integer 75 78 81 83 87 Interactive 62 64 IntFormat 90 IsAlNum 86 IsAlpha 86 IsDigit 86 IsKeyDown 46 IsLeftButtonDown 48 IsRightButtonDown 48 IString 90 164 it 25 item 29 30 label 39 65 LARGE 25 Large 25 large 25 LeftMargin 23 Length 101 LINE_STR
98. ens up by left clicking Editor Then the user interface depicted in Figure 1 2 is displayed 14 1 6 SigMath documents Fig 1 2 Editor window In this editor SigMath documents can be created and compiled Table 1 1 lists the menu points and their functions The editor provides keyboard shortcuts for improving work flow Shortcuts are key combinations connected to frequently used functions They are listed in Table 1 2 Another important feature of the editor is that it allows to place anchors or marks in the text Anchors are labels on lines which can be referenced to If one is working on a document containing a lot of source code and one has to modify a certain text passage repeatedly one can use an anchor to jump to this text passage Anchors are placed by using the key combination CTRL Shift Number Any number between 0 and 9 is possible The number appears then on the left margin of the line To jump to this anchor one uses CTRL Number To remove the anchor move the cursor to the corresponding line e g by clicking on the line and hit CTRL Shift Number The editor quits by hitting ESC and one returns to SigMath s start page 1 6 SigMath documents 1 6 1 Basic structure We illustrate the basic structure of SigMath documents by a simple example Let us open the text editor in SigMath s user interface Every document consists of a preamble and a main part The preamble may contain the definition of globally
99. ent of the table is written sequentially in the code To place text in the next column one enters amp and a new line is started by If we want to add a horizontal line we put a hline before the next line break Syntax hline One difference to the usage in ATEX is the following in SigMath the number of columns does 33 2 Word processing Fig 2 4 Example for a color gradient not have to be specified beforehand That means the token amp acts as a tabulator in ordinary word processing sofware Where the next tabulator is set depends on the longest width of the text in the same column There is no automatic line break within a column and every row can be ended at any time since no predefined number of columns has to be filled Spacing Using the properties ArrayColSep and ArrayStretch the column separation distance and line space can be adjusted respectively Syntax ArrayColSep ArrayStretch Their value is 1 0 by default The column separation distance is given by ArrayColSep mul tiplied by the width of a blank in the current font size The line space is given by ArrayStretch times the current font height To get the current value of the column speration distance we can use the ColSepDelta command Syntax ColSepDelta This command is read only i e we cannot alter the value of the separation distance in this way Finally all distance modifications are active for the current table i e for the whole array 34
100. ent using PlotIndex To change properties of a plot or of a certain plot item one should save the corresponding indices to variables Then BindPlot and BindPlotItem can be used to select a plot or plot item in order to change some of its properties Selecting a plot item the corresponding plot is activated Syntax BindPlot index BindPlotItem index After that command all property of the plot can be modified We can change the color of the background adjust line patters or request axis areas of the active object see Section 4 1 4 With PlotReDraw we force SigMath to redraw the plot using the new properties By default a plot is redrawn automatically if one variable used in the depicted function is changed by user action Also if the version number of an array changes see p 95 in Section 6 3 2 the plot is redrawn automatically 57 4 Plots in SigMath We provide an example to summarize this description Example 4 3 We want to generate and draw a simple combined plot Some objects shall be made adjustable by clicking several interactive elements Compiling the file BindPlot txt a white colored circle on blue background is displayed see Figure 4 4 In this example the circle is generated using two semicircles 2 plot items With the elements at the bottom of the screen we can change the color of the upper or lower semicircle or background color To change the color of one of the semicircles we have to activate the corr
101. ently implemented environments in SigMath in Table 2 1 2 2 Page layout In this section we introduce how to layout pages in SigMath at first without dealing with content that appears on the page later on 21 2 Word processing The simplest page layout we encoutered was a blank page with white text on black background see Section 1 6 1 Some predefined simple page layouts can easily be used by inserting the corresponding layout template which can be found in the menu item Layout of the Template Name of the environment Description Reference array Tables and matrices Section 2 9 centerline Centering text Section 2 4 document document environment Section 1 6 1 enumerate Listings Section 2 6 env environments for internal use Page 68 inputlabel formatted text in links Page 40 plot2d 2D graphics Section 4 1 plot3d 3D graphics Section 4 2 postext Positioning of text Section 2 5 Tbl 2 1 Environments in SigMath menu of the text editor For example on left clicking the template Standard with date multipage code is added to our possibly empty document To save and compile the document select Compile the page layout as depicted in Figure 2 1 is displayed 22 Text on the left PAS SO 5 AO ALO Text on the right Sal Fig 2 1 The page layout template Standard with date multipage 2 3 Fonts Let us have a look at the code lines that were adde
102. er The method clear deletes all elements of a container and leaves the container empty The method returns the data type Void Syntax S sizeQ S clear AS emptyO Example 6 2 Let us have a look at the effect of these methods 104 6 4 Control structures variable S SetQ S insert 5 variable i S size S clear S empty The first line generates a new container S of type Set Then we add the element 5 using insert The query S size returns 1 and assigns this value to the Integer variable i In the following S clear empties the container which results in the answer true in the last line S empty For the user all iterators have produce the same effect on all types of containers Internally every container type implements its own iterator method The methods begin and end provide iterators which point to the beginning and the end of a container For the resulting value of the method the actual type of the container does not take effect Only the way the result is generated depends on the type The methods pop_front and pop_back allow to delete the first and last element of a con tainer respectively The methods push_front and push_back allow to move the data element from its current position given by the iterator to the beginning or to the end of the container We can compare container variables using and Two containers are equal if they contain the same elements in the same order Otherwise th
103. er s Here the first two commands act on the String S directly The last two functions leave S unchanged and return a modified String which has to be stored to another variable The Trim method removes all leading and closing blanks and all control symbols Syntax S Trim string Trim s The first command directly acts on S whereas the second variant works as a function and returns the modified string Methods to export strings This paragraph contains methods to export and import text To import text from an existing text file we can use the method Load Syntax S Load filename Here filename is the full qualified path name of the file to be loaded drive directory and file name also see Table 1 3 The method returns false if the file is not found or cannot be loaded The counterpart of Load is Save It allows to save the value of S to a text file Syntax S Save filename Here filename is an expression of type String which specifies the filename and target path of the file which afterwards contains the value of S If we want to load only parts of a file or save more than one string to a text file we have to use the following two methods instead Syntax S SaveToKey filename key leadingString S LoadFromKey filename key leadingString The SaveToKey method can be used to save the value of a string variable to a certain marked position a key in a file Here filename is the name of a file
104. ermine how in relation to the previously defined origin the text block is aligned Let us assume for the moment that the textblock consists of a single line only Virtually we draw a rectangle around the text Its width equals the width of the text block and its height is given by the font size that is valid in front of the current text block In local coordinates the center of the rectangle has coordinates 0 0 the lower left corner is specified by 1 1 and the upper right corner is given by 1 1 Now textX and textY specify which point in local coordinates is mapped to the origin For text blocks containing more than one line the defining rectangle is only adjusted to the first line Finally the content of the text block is given by a postext environment There are two ways to specify a postext environment Syntax postext here you put text commands begin postext text or commands end postext Example 2 1 Let us study the effect of text positioning on the text block Sample text The center of the screen is our target point and we use the following text coordinates 1 TextPos 0 5 0 5 0 0 2 TextPos 0 5 0 5 1 1 3 TextPos 0 5 0 5 1 0 28 2 6 Enumerations 4 TextPos 0 5 0 5 1 5 2 Note the placement of the descender of the letter p especially in the second example The results are depicted in Figure 2 3 a TextPos 0 5 0 5 0 0 TextPos 0 5 0 5 1 1 ee O MN TextPos 0 5 0
105. erties and methods the class provides The elementary components of a variable of complex data type are called data fields and can be interpreted as variables of elementary data type Methods data fields or properties of an instance of the class are called using dot notation see Section 6 1 and thus do not require the variable as an input argument There even exist methods of the same name their uniqueness is guaranteed by class membership of the calling object in front of the dot SigMath provides the following complex data types array cont glpoint string bitmap and edit Arrays provide the possibility to store homogeneous data in a vector a matrix or some higher dimensional object Containers are generalized list structures with additional sorting properties The GLPoint data type stores coordinates in three dimensional space and allows to easily describe movements or changes of viewpoint by a fourth dimension The String data type already introduced in Section 6 2 3 is also a complex data type It allows to modify sequences of characters far beyond the elementary concepts We will provide more detail on all complex data types and describe their methods in the following sections 6 3 1 Strings The String data type is a storage template for sequences of characters or shortly text Since text may have arbitrary length we cannot allocate a fixed amount of space for such a variable beforehand and thus sequences of characters require
106. esponding plot item To change background color the whole plot has to be activated Code BindPlot txt function f x function g x return sqrt 1 x x variable PI 0 Plot index variable PII1 O PlotItem index variable PII2 0 variable size 0 68 return sqrt 1 x x function UpperCircleRed BindPlotItem PI1 PlotColor 1 0 0 PlotRedraw function UpperCircleWhite BindPlotItem PI1 PlotColor 1 1 1 PlotRedraw function BgGreen BindPlot PI PlotBgColor 0 1 0 PlotRedraw function BgBlue BindPlot PI PlotBgColor 0 0 1 PlotRedraw 58 4 1 Two dimensional plots begin document Below this text a circle is shown whose properties can be changed using the buttons at the bottom begin centerline begin plot2d PI PlotIndex PlotWidth size anim Width anim Height PlotHeight size PlotAxes PlotBgStyle FILL XPlotBgColor 0 0 1 XPI1 Plot f 1 1 PI2 Plot g 1 1 end plot2d end centerline DoString upper semicircle in red UpperCircleRed DoString upper semicircle in white UpperCircleWhite DoString background color green BgGreen DoString background color blue BgBlue end document a Start b after clicking upper semicircle in red c after clicking background color green d after clicking upper semicircle in white Fig 4 4 Redra
107. eterPages txt As soon as the user reaches this page the SigMath graphics export GUI appears We provide a screenshot of this GUI in Figure 5 1 If the document consists of one page only put the above line directly following begin document If we have a multipage document structured by newpage label label0fPage we insert the line directly after the newpage command All menu items are listed in Table 5 1 In the following pages we introduce the available options We can specify the target format font families of text in figures sort and coding methods and further general options Some file formats allow to save additional title and producer of the file Format The most important choice you have to make in exporting graphics is the format you want to save it to This choice affects which applications can open the saved file and the 65 5 Exporting graphics Courier default ExportIDE txt Fig 5 1 The graphics export GUI Menu item Function Draw generates the figure without export Export as opens the Save as dialog window then enter the file name Export as and open saves the figure as in Export as and opens the default ap plication for the corresponding file format Help provides help and further information on the options About provides additional information Tbl 5 1 Menu items of the graphics export dialog window possibilities of post processing the data Al
108. exist during execution of the code but not simultaneously 6 2 3 Elementary data types Every data types comes along with specified memory templates This template organizes how much memory bytes is allocated and how to interpret it The number of bytes determines the range of the variables interpretation of 0 and 1 gives a meaning Thus a 1 Bit is interpreted as true if the associated variable is of type Bool in case of an Integer variable the interpretation would be the number 1 Besides there are data type specific operators Elementary data types are the building blocks for higher more complex data types see Section 6 3 Arithmetical data types The arithmetical data types are given by Integer Real and Complex The range of an Integer is a subrange of all integers whereas data of type Real are floating point numbers The set of all floating point numbers is a finite subset of the set of all rational numbers The data type Complex represents the complex numbers and contains the constant I for the imaginary unit Any value of type Complex is stored as a pair of two floating point numbers The notation for constants is given by c a b I where a and b are of arithmetical data type By using the complex unit 1 SigMath interprets this as a complex number Mind that even if a complex number c is specified in the form c 5 2x l it is treated as a pair of floating point numbers and not as a pair of Integer values Table 6 5 gives the ran
109. f text TEX users will recognize a lot of the features in SigMath In fact they were in the focus of the developers of SigMath Nevertheless the experienced TFX user should be aware of some subtilities that arise from the way SigMath presents text If you want to write several documents we recommend to generate one file only containing the layout specifications and include this file to the preamble of all text documents using the Include command see section 1 6 3 Furthermore layout specifications only apply to the current i e the environment they are contained in see section 2 1 within 2 1 Environments Before we treat more concise details of the word processing features of SigMath let us introduce the abstract notion of an enviroment Environments are used to group several commands or text and to separate it from the rest of the document As known from e g ATEX enviroments are given in the following form Syntax begin Name0fEnvironment some text or commands end NameOfEnvironment Everything within such an environment is processed according to certain specified parameters Thuis text within the enviroments may appear in different color font size or font family than other text in the document Very simple enviroments are given in the following form text outside Y newTextColor text inside the environment text outside In this example the text outside appears in another text color as the text inside We list all curr
110. f the if else structure false Boolean constant for part of the for loop function definition of a function I imaginary unit if part of the if else structure INF constant infinity NAN constant for Not A Number Pi Pi 3 14159 switch part of the switch case structure true Boolean constant use_class specifies a certain class variable definition of a variable while part of the while loop Tbl 6 4 Reserved words in SigMath The name of a variable can have arbitrary many characters and is case sensitive Although there are basically no limitations to our creativity we suggest to use speaking names and care about uniqueness One should not define two variables having the same name Only in very limited cases as e g for the counting variable in a loop see Section 6 4 2 one can use the same name for different locally valid variables in different environments Apart from uniqueness and meaning one should be consistent Names of variables or func tions in this guide always start with an uppercase letter Several words in one name are easily separated by using upper case letters MoreThanOneWord Only names consisting of one char acter only are excluded like e g x Arguments in function heads begin with lower case letters formalArgument Thus the way a name is spelled already contains information on the mean ing and usage of the variable To determine the data type of a given variable we use the Type0f func
111. fault 156 8 2 Writing DLLs break return DLL_RESULT_OK After setting a pointer F by calling GetFunction one can use the following function int F int argType int argSize void argPointer The arguments have the same meaning as at the beginning of this section If argPointer contains one entry more than the associated user defined functions the last entry is used as return value The following data types are supported Arguments Real RealArray Integer String Return value Real RealArray Integer Example 8 4 Let us consider the following example We want to use the function function f x return sin x in our DLL We write RECALL_FUNCTION F 0 char name f F GetFunction name double x 1 0 y 0 0 int ArgTypel 4 DLL_ARG_DOUBLE 0 DLL_ARG_DOUBLE O DLL_ARG_NONE void ArgPointer void amp x void amp y F ArgType 0 ArgPointer FreeFunction F Then y sin 1 157 9 Rendering 159 Index 85 94 80 81 18 81 82 80 81 85 81 95 81 82 85 INF 92 80 81 81 82 INF 92 80 81 18 18 81 82 lt 81 lt 81 81 81 105 gt 81 gt 8l 18 amp 33 amp amp 81 83 80 81 gl LineWidth 129 gl PolygonStipple 131 gl Rotate 136 gl Scale 135 gl Translate 136 17 27 27 27 Ne 20 or 26 29 33 8105 77 Ang
112. gStyle NONE PlotBgStyle RECT PlotBgStyle FILL 56 4 1 Two dimensional plots XPlotBgColor r g b The option NONE gives an empty background and no frame RECT generates a frame around the plot window and FILL allows to fill the whole plot window If we have chosen one of the last two properties we can adjust the color of the frame and the background using the property PlotBgColor in the usual RGB notation For other background designs we can put a wallpaper in the background using the Image command see Section 2 7 Make sure to specify an appropriate value for the depth of the image Basically the whole plot window is used to draw the plot If we want to add margins we can specify its corresponding sizes by using PlotMargins Syntax XPlotMargins left right top bottom PlotMargins all All parameters are numbers which specify the size of the margin using relative length with respect to the plot window Using the first variant the size of the four margins is specified separately The second variant treats all four margins the same The default setting is given by PlotMargins 0 1 0 1 0 1 0 1 or equivalently PlotMargins 0 1 i e a tenth of the width and height is left for the margin 4 1 3 Redrawing plots Let us continue with the considerations on plot referencing We already learned that every plot item has an identifier its index The index of a plot can be accessed within the corresponding plot2d environm
113. ge of the data types and the required memory sizes Data type Range Memory size Integer 2147483648 to 2147483647 4 bytes Real 1 79769 10 308 8 bytes Complex 1 79769 10 308 1 79769 104308 16 bytes Tbl 6 5 Ranges for variables of type Integer Real and Complex Boolean variables Boolean values have the identifier Bool in SigMath They can only take the following two values true or false The data type Bool often appears as the result of an expression of the form 5 gt 10 Typically we use boolean values in connection with if conditions see Section 6 4 1 78 6 2 Data types Void the empty data type Variables of type Void can not be defined explicitly in SigMath This data type is only used internally to simulate a return value of functions that are not returning any value If one writes a function that should change the value of a global variable we do not need a return value Such functions implicitly define a return value of type Void Pointer The data type Pointer is used to store memory addresses It is only used in connection with DLLs see chapter 8 To define a variable of type Pointer we write Pointer Syntax variable P Pointer variable Q P Both commands generate a Pointer which points to the zero address The comparison operators and are also available for Pointer Also comparison of Pointer and Integer variables are poss
114. he correct position is determined internally The method ToString is used to convert the given container variable to a String and returns the result This function uses the iterator i as a loop variable of the for loop and thus runs through the set S The method begin returns the iterator on the first element of the container This is set to the next element by applying the 103 6 Programming increment operator until we reach the end of the container The last element of a container is addressed by end In between the method ValueAt which needs the iterator as its ar gument reads the value at the corresponding position Finally the compound assignment generates the resulting String Code Xfunction ToString cont S variable res for variable i S begin i S end i res ValueAt i return res begin document variable S Set S insert This is S insert the way S insert to sort S insert Strings S insert variable Display ToString S ValueOf Display end document Methods for containers In this section we introduce some of the elementary methods provided by the class cont for containers To detect the number of elements in a container SigMath provides the method size If one wants to decide if a container is empty we suggest to use empty The result is of type Bool and has the value true if there are no elements in the contain
115. he colors of the corners are specified anti clockwise starting at the bottom left corner using their RBG values and alpha value That means the argument of BGColoredRect 32 2 9 Tables Command Description Color text color GetColor BgColor background color GetBgColor GoToColor color of text in links GetGoToColor InputColor color of text in inactive input fields GetInputColor InputColorFrame color of the dashed frame around active input field GetInputColorFrame InputColorSelText color of selected text in active input fields GetInputColorSelText InputColorSelBg text background color of selected text in GetInputColorSelBg active input fields Tbl 2 4 Commands for colors consist of a comma separated list of 16 entries with values between 0 and 1 Example 2 5 To generate a color gradient as depicted in Figure 2 4 use the following code BgColoredRect 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 2 9 Tables Tables can be generated in SigMath by using an array environment Please do not confuse the array environment with the complex data type array which will be introduced in Section 6 3 2 All commands concerned with the display of such variables will be introduced there Basically speaking a table is just a special arrangement of ordinary text and thus all previ ously mentioned text layouting commands and properties like font family and font size carry through As known from TFX the cont
116. he polynomial at the end of the calculations The functions f x and fDerivative x calculate the function value and the value of the derivative at the point x The iteration is implemented in the function Newton x which is called from the document block and contains a variable to store the returning value 118 6 7 Debugging Finally using BreakPoint we can monitor n The values of x and FunctionValue are displayed after every call of the function Newton x Code Newton txt Xvariable x 3 5 function f x return x 3 6xx 2 4 x 24 function fDerivative x return 3 x 2 12 x 4 function Newton x return x f x fDerivative x begin document variable FunctionValue 0 0 BreakPoint Iteration x Newton x FunctionValue f x end document The first interruption of execution is performed after the first call of Newton x Thus x was already updated and contains the value 1 27 During compilation BreakPoint has no effect SigMath just gives a warning message All information is saved to the file ProgDir Shared Temp BreakPoint txt In order to compile and display the content of this file in SigMath one has to load ProgDir Shared GUI FormatBreakPoint txt 119 7 Computer graphics This chapter should be seen as a starting point for OpenGL programming We start with a short introduction to OpenGL and then consider the building blocks for every complex graphical object
117. here generates a sphere from quadragons and triangles Syntax glu Sphere radius longitudinalScaling latitudinalScaling The model for the sphere is the globe The poles are automatically generated and the connect ing line runs along the z axis of the global coordinate system This way longitude and latitude is defined At the poles the surface of the sphere is constructed from triangles in between from quadragons In summary the sphere consists of longitudinalScalingxlatitudinalScaling pieces Cylinders The function glu Cylinder allows to draw cones Syntax glu Cylinder radiusBottom radiusTop height horizontalScaling verticalScaling The cone rests on the xy plane of the global coordinate system and the top is pointing in the direction of negative z axis The horizontalScaling gives the resolution around the z axis the verticalScaling the resolution along the z axis One can draw cylinders and pointed cones depending on radiusBottom radiusTop and radiusTop 0 respectively Disks and disk segments A disk is a round surface having a hole in the middle like a CD They can be drawn using glu Disk Syntax glu Disk innerRadius outerRadius horizontalScaling radialScaling The arguments innerRadius and outerRadius determine the radii of the hole and the disk respectively The center of the disk is located at the origin of the global coordinate system 132 7 2 Primitive and this disk lies in the xy plane Th
118. how to load and map textures We focus on a hands on introduction to textures and will not go into the theoretical details 7 7 2 Types of textures Textures can be classified based on their dimension There are one two and three dimensional textures Here we focus on two dimensional textures Besides discrete textures where images are used as textures there also exist procedural textures In this case the structure is generated 141 7 Computer graphics from a mathematical algorithm Discrete textures are easy to use A digital camera and image processing software allows to generate as much images as we want Memory consumption is larger than for procedural textures though On the other hand generating procedural textures may get very complicated Basically one rather uses procedural textures in regular structures and discrete textures for irregular detailed structures 7 7 3 Loading textures In order to use a texture we have to load it to video memory If the available memory is small data has to be stored in main memory and results in slowing down graphical display Discrete and procedural textures can be implemented in SigMath in a very similar way We explain the procedures for discrete textures in detail and restrict to pointing out the difference in the case of procedural textures Discrete textures In the text editor you can find several templates to load textures We will have a look at two templates TextureLoad and
119. ible One can use P to test if P directs to the zero address Strings Since variables of type String are fundamental objects in SigMath s programming language we introduce them in this section on elementary data types Nevertheless it is in fact a complex data type We will come back to its interpretation as a class in Section 6 3 1 In contrast to arithmetical data types a value of type String is used to present sequences of characters of arbitrary length Those sequences of characters are delimited by quotation marks Since those delimiters may enclose character sequences of arbitrary length the data type String allows easy manipulation of text Let us give some examples for the initialization of variables of type String variable S variable T Hello variable V A short text with blanks Since the length of a String variable can change one cannot allocate a fixed amount of memory beforehand Its memory consumption is dynamically adjusted to the value of the variable i e changing the value during run time changes the amount of reserved memory 79 6 Programming 6 2 4 Operators and expressions To handle variables according to their data types we use operators or functions see Section 6 5 Operators are predefined functions which allow for a different syntax The arguments are typi cally not given after the name of the operator in parenthesis the operator is written as a short symbol and in ca
120. in the second line e g by y x 5 since x is not used later on and the current value need not be stored The animation oriented interpretation in SigMath leads to the following in every frame 5 is added to x since almost all source code is processed in an infinite loop The displayed value on the screen increased more and more without any user interaction That is not what the program was intended to do This little example should give you an idea that not all familiar programming concepts lead to the desired results This includes self referencing commands as in the example that should at least be handled with care The advantages are clear the programmer does not have to deal 12 1 3 Installation with repetitions of his source code and can even make use the effect shown in the pseudo example 1 3 Installation SigMath comes as a zip archive In order to install a new version of SigMath remove an older existing version by deleting the SigMath program folder before unzipping the new version The following instructions are valid for the operating systems WINDOWS 98 2000 NT XP and WINDOWS VISTA 1 Download the file SigMathZIP exe from the SigMath web page http www math tu dresden de sasvari sigmath 2 Start the executable file by double clicking it You are asked for the target directory Choose a path and click on Extract 3 Open Explorer choose the SigMath directory in the path and right click the file SigMath
121. ion Thus they may hide global variables To use a function one does not have to access its source code but only needs to know the class or document it is defined in and its interface its name and a list of arguments given in parenthesis Finally we need to know the data type of its return value to use the function properly Function definition Before we can use a function we have to define the function using the following syntax Syntax function functionName argument1 argument2 112 6 5 Functions commands return ReturnValue commands The first line is the function header The reserved word function begins the definition of a function Its name allows us to call the function at any position of the document The name of a function within one block has to be unique even if the functions would have different lists of arguments Only the name of the function is used to identify the function SigMath will execute the function if the given list of arguments in the function call matches the list of arguments given in the definition The list of arguments may contain zero or more arguments argument1 argument2 are identifiers according to the naming conventions of variables Within the function the arguments can be used as normal variables They are initialized on function call according to the assigned values Additional definitions of the arguments are not possible Dif ferent arguments must have different identifie
122. ionally one can provide the alpha value as a fourth component By default its value is 1 It codes opaqueness of a given object and takes effect if several objects overlap Using an alpha value of 1 means opaqueness or 0 transparency Image we look at a red ball behind a green colored pane If the pane has an alpha value of 1 the red ball behind is not visible The lower the alpha value of the window pane the more of the ball is shining through Finally an alpha value of 0 results in a transparent window pane With this basic principle in mind we can color different objects using one of the commands listed in Table 2 4 followed by the three color components as arguments Syntax ColorCommand r g b alpha For example using Color 1 154 text color is set to white Every command that sets a color is accompanied by a command to return the current color of an object The commands are simply given by adding the prefix Get to the command Syntax GetColorCommand For background design two further commands are available BgImage allows to use an image as a wallpaper The command behaves as the Image command in Section 2 7 As for the Image command the positioning and scaling of images is controlled by the ImageStretch property The second command BgColoredRect produces smooth color gradients starting from specified colors in the corners of the screen Syntax BgColoredRect bottomleft bottomright topleft topright In this command t
123. ipboard Here LoadFromClipboard writes the whole content of the clipboard to the variable S whereas SaveToClipBoard writes the value of S to the clipboard Methods for data type conversion The following section introduces functions that are not acting on String variables but are com mands provided by the string class An important group of commands is devoted to converting variables of different data types to String and the vice versa To convert variables to String variables we use the ToString function It automatically de tects the data type of its argument and performs the conversion based on certain properties Syntax string ToString X The argument X is a variable we want to convert The variable has to be of arithmeti cal boolean of GLPoint data type For arithmetical variables we can specify by the property PlusString which text sign has to be placed in front of the number if it is positive The text 89 6 Programming sign has to be specified in between to vertical lines Syntax string PlusString l By default no sign is written for positive numbers For floating point numbers the FloagDigits property allows to specify the number of decimal digits that are displayed Syntax string FloatDigits Apart from that there are several different representations for floating point numbers The property FloatFormat allows to choose between three different types Syntax string FloatFormat string FIXED_POI
124. is can be quite useful if we want to write some text onto the surface of an object Syntax bmp TextureFromString text fontBase w h The function TextureFromString in the class bmp returns the ID of a texture that contains the String text The argument fontBase is of type Integer and specifies which font should be used Here 0 uses the current font w and h determine width and height of the texture The texture matrix We already encountered modelview and projection matrices in connection with transformations Before objects are displayed the object coordinates are multiplied by the modelview matrix Similarly the texture coordinates r s t q are multiplied by the texture matrix before they are drawn This allows to scale translate and rotate textures 146 7 7 Textures Remark Since we only consider two dimensional textures the coordinates r and q can be neglected Thus we always assume them to be 0 7 7 4 Mipmapping INSERT 7 7 5 Texture mapping Texture mapping denotes the process of mapping texture to an object That means that the coordinates s and t of the texture are mapped to coordinates x and y of the object We have already seen that the values of texture parameters see Section 7 7 3 have an effect on this procedure In the following we will have a look at this mapping Texturing primitives On the level of primitives texture coordinates are mapped pointwise to the vertices of the prim itives Before we c
125. item of level 0 nextlevel item level 1 nextlevel item level 2 second line of the last item in level 2 prevlevel item leveli prevlevel item item 3 of level 0 end enumerate 2 7 Inserting figures As we will see in forthcoming chapters SigMath offers multiple possibilities to generate ani mated figures on its own Nevertheless already existing figures can be embedded into a Sig Math document The command for inserting figures is the Image command It generates a rectangle of user specified height and width and screen position that contains the figure The Image command then loads the file and transforms its content to an object a texture which can be displayed by SigMath If the figure does not change from frame to frame it is not reloaded Nevertheless SigMath checkes this condition at any frame and loads a new figure if necessary see the online help for an example Furthermore one can control how the figure is placed within its rectangle display it using its own width and height or maybe distort it to fill the whole rectangle The corresponding behavior is adjusted by the ImageStretch property Its value takes effect on the following Image commands in the document Syntax ImageStretch FILL ImageStretch PROP ImageStretch PROP_MAX_2 Image name left top width height depth On FILL the figure fills the whole rectangle and might be distorted PROP rescales the figure preserving aspect ration
126. ith the GoToPage command one can display formatted text using an inputlabel environment See page 40 for a description The second argument fctName contains the name of the function which is executed The function must not have any arguments It is given in quotation marks without parenthesis The second version contains the argument ident which can be used to pass the information to the function which DoString command called it Without the third argument ident is automatically set to text A small modification of the DoString command is given by the DoValue0f command Syntax DoValueOf varName fctName DoValueOf varName fctName ident As before clicking invokes a function It displays the value of the variable varName also com pare the behavior of ValueOf in Section 6 2 2 As was mentioned before an identifier ident is given implicitly or explicitely which can be used to detect what kind of link the function was invoked of Thus several different links can invoke the same function executing different commands In the function the value of ident can be requested by using SelectedDo Syntax SelectedDo Example 3 1 As an example we want to change displayed text of the link to the value of its identifier variable S Al 3 User interaction function f S SelectedDo XDoString Text1 f XDoString Text2 f Last function ValueOf S After compilation there is no text following the col
127. ition in the loop header is no longer satisfied To abort the loop one can add a break command Depending on another condition inside the loop body one can stop further repetitions In case of a break SigMath resumes execution at the first command after the loop If one wants to stop the execu tion of the current repetition only one uses continue Then the remaining commands in the loop body are skipped and a new repetition starts Example 6 7 Let us consider the following question Given a small non negative real number e e g e 0 0264 we want to compute the smallest integer 1 100 such that 1 i lt One naive idea is to use a for loop with counting variable i Code variable epsilon 0 0264 variable i 1 for i 1 i lt 100 i if 1 0 i gt epsilon continue else Display i is WalueDf i break As long as i is not large enough we directly restart the repetition using continue This is done until either i 100 or we the condition is satisfied In this case the value of i is displayed and the for loop is aborted by break While loops Another possibility to repeat the execution of a code segment is given by while loops The loop header contains the while command and a condition Initialization has to be performed before hand and the reinitialization is part of the loop body Thus there are basically no limitations We can e g use more than one loop variable 111 6 Programming
128. ixel is colored if the texture is larger than the surface The values gl LINEAR and gl NEAREST have the same effect as in the magnification filter More information on the other possible values can be found in Section 7 7 4 on the mipmapping technique gl TEXTURE_WRAP_S This parameter controls if the texture shall be repeated horizontally This way of repetition is also called tiling Either we enable repetition gl REPEAT or disable repetition g1 CLAMP Repetition is enabled by default 143 7 Computer graphics gl TEXTURE_WRAP_T This parameter controls vertical tiling and behaves like the horizontal case After adjusting all texture parameters we have to bind the texture variable to the image in a file Its file name is the argument to the function TextureLoad Finally glu Build2DMipmaps loads the image to video memory Syntax glu Build2DMipmaps filename Width and height of the image measured in pixels have to be equal to a power of two If this is not the case the command automatically scales the image to the next larger admissible size Alternatively we can specify a bitmap as argument in the function glu Build2DMipMaps instead Syntax glu Build2DMipmaps image Finally gl BindTexture gl TEXTURE_2D 0 removes the bind of target and texture and return TextureNr returns the texture ID as return value of the function The texture ID 0 is reserved and represents an initial texture All targets are bound to
129. k of writing a one or two dimensional table as an array environment see p 35 to an array variable one uses the InputArrayBlock command which has been introduced in Section 3 3 97 6 Programming Methods to manipulate arrays After filling the array and showing it we want to introduce means to change the values of an array All methods work on the corresponding array directly That means previous layout may be overwritten and get lost Also note that some of the following methods are only available for one or two dimensional arrays First we can rearrange an already existing array to obtain some higher dimensional array Syntax NA Resize n0 nd A ResizeInit n0 nd defaultValue The first line transforms A to an array of layout n0x xnd If this increases the size of the array missing values are filled up with null elements of the corresponding data type The second variant can be used to change the default behavior All missing values are automatically filled up by defaultValue variable I IntegerArray 0 I FillFromString 1 2 3 45 67 8 I Resize 2 2 2 In this example we generate a one dimensional array i e a vector of length 8 and rearrange it afterwards to an three dimensional array with layout 2 x 2 x 2 The rearrangement follows the sequential storage pattern we introduced on p 94 The result of our transformation reads as 1 0 0 0 1 1 0 0 1 2 1 0 1 0 3 1 0 1 1 4 1 1 0 0 5 1
130. l available graphic formats are given in Table 5 2 If one does plan to insert the figure into further documents POSTSCRIPT or PDF are the for mats of choice For usage in webpages one should preferize SVG files since most web browsers can display those embedded graphics directly or via plugins Graphic import in ATEX documents needs a little bit more explanation The easiest way is to export to EPS format and directly include it into the ATEX document To do this one has to load the corresponding packages in the preamble of the TFX file usepackage dvips graphicx Next one includes the figure by 66 File format Description ps POSTSCRIPT eps Encapsulated POSTSCRIPT POSTSCRIPT with further fea tures needed for e g ATEX pdf Portable Document File svg scalable vector graphics in XML syntax suitable for usage in web pages pet Portable ATEX Graphics Format tex generates a ATRX file Tbl 5 2 Supported file formats in the graphics export begin figure parameter for positioning includegraphics parameter on figure size Filename eps caption caption label label for later reference end figure This is a fast way to export SigMath graphics and include them to ATEX documents Another option is to leave text formatting to ATEX To do this change the option No text in Op tions to On The code remains unchanged Next we generate an EPS file by specifying the cor
131. le and intuitive one can easily add interaction to word processing and programming task Especially all of SigMath s functionality around generating animations benefits from the possibility to interact with the user 3 1 Hyperlinks Hyperlinks or links are elements on a SigMath document that are connected to an action The action is executed every time the user selects the link by clicking on the element Typically links refer to different pages or documents as links in HTML documents or execute certain functions Links to pages The simplest form of an interactive element is a link to a different page A page in SigMath is the displayed content of a SigMath document we can navigate through by using arrow keys or scrolling with the mouse Pages may contain text and graphical objects whose generating commands are contained in the document block of a text file or are embedded by using Include commands see Section 1 6 3 To generate new text or code that should be displayed on another page we separate both pages by using the newpage command This command can be interpreted as the SigMath analog of page breaks Syntax newpage With this command a new page is generated within the current document that contains all content following newpage in the source To make the page accessible from other pages we have to assign page labels by using label Syntax label PageLabel Labels can consist of letters and numbers After that one
132. leDegree 101 AngleRadian 101 anim 72 ansi 24 ANSIToString 91 arabic 24 array 22 33 35 44 84 113 ArrayBlock 35 97 ArrayColSep 34 ArrayColX 36 ArrayGetCell 36 37 ArrayIndex 36 ArrayRect 36 ArrayRowY 36 Arrays 75 ArrayStretch 34 AutoTicks 60 64 AxesColor 60 64 AxesDepth 61 64 AxesRanges 60 AxisMaxRange 62 AxisPos 61 AxisRange 60 63 back 94 Backslash 85 BaseLineStretch 26 begin 105 begin document 16 bf 25 161 Index BgColor 33 BgColoredRect 32 Bglmage 32 BINARY 91 BindPlot 57 BindPlotItem 57 Bitmap 75 145 bitmap 84 bmp LoadPower2 145 Bool 75 78 Brackets 35 break 108 111 BreakPoint 118 Busywaiting 73 case 76 107 108 centerline 22 26 chinese 24 clear 104 code 17 20 77 ColAlign 35 Color 32 33 124 ColSepDelta 34 Complex 75 78 82 90 cont 84 113 Container 75 continue 111 crArrow 48 crCross 48 crDrag 48 crHandPoint 48 crHourGlass 48 crNo 48 CrossProd 101 crSizeNESW 48 crSizeNS 48 crSizeNWSE 48 crSizeWE 48 crUpArrow 48 CurrentPageLabel 40 CurrentPageNr 40 Cursor 48 DECIMAL 91 162 default 24 76 108 Delete 87 Digits 42 61 Dim 95 discrete textures 141 DocDir 18 19 DocName 18 19 document 22 DoString 41 109 DoValueOf 41 e 76 edit 84 else 76 106 108 empty 95 104 end 105 end document 16 endif 107 enumerate 22 29 env 22 68 environment 21 Execute 20
133. lements_d Here ArrayName is a valid name for a variable see p 75 for the rules to generate variable names Array Type is one of the array types listed in Table 6 10 Once this type is specified all elements have to be of the corresponding data type Number_Elements_1 Number_Elements_d is a list of nonnegative integers that specify the number of elements that can be stored in the corresponding dimensions of the array If the list is empty i e d 0 the array cannot be used to store data Nevertheless the variable exists and all methods can be called The total number of elements in an array called size of the array is implicitly determined by the definition It is given as the product of all numbers of elements in the dimensions Size Number_Elements_1 x Number_Elements_2 x x Number_Elements_d The size of an array is zero as soon as one of the sizes of its dimensions is zero Thus arrays can have size zero although they have at least one dimension The layout of an array is given by its dimensionality and the number of elements in each dimension In SigMath there are no limitations on the size or layout of an array i e up to restrictions of physical memory arrays can take arbitrary sizes Samples of array definitions are given as follows variable M IntegerArray 3 4 variable N RealArray 2 4 0 This way Mis an Integer valued 3 x 4 matrix i e the first dimension contains 3 and the second dimension
134. line of the document Home Jump to beginning of current line CTRL Home Jump to the first line Tbl 1 2 Shortcuts in the editor Code HelloWorld txt begin document Hello World end document 1 6 2 Text commands and comments If we only insert plain text in the document it will be displayed in white letters using a de fault font RM and font size small on a black screen The actual size depends on the size of the monitor and the specified graphics resolution Commands can be used at any point in code Commands are denoted by and thereby detected by the interpreter the compiler of the source code in a SigMath document Several commands can be grouped using the code command code here arbitrary many commands without using Everything contained within the curly brackets is regarded as a command so has to be omitted To improve readability of code one should include comments to the source code It can be used to explain the meaning of certain variables commands or even of a part of code This helps other programmers to understand what you want to do or even helps yourself if you want to understand what you intended to do when writing the code In SigMath we have to distinguish comments within the text regions and the command regions of the document Command regions are e g code blocks or functions Comments in the text 17 1 Introduction may be embedded as in ATFX by using This character
135. llowing form Syntax int PrefixDLLPROP_indentifier_ReturnType int argTypel 1 int argSize void argPointer Here PrefixDLL is a predefined prefix By default the underscore _ is used The suffix Re turnType represents the data type of the returned value of the function or the data type of the constant in SigMath All possible suffices are summarized in Table 8 1 Both DLL functions return a value i of type Integer If i 0 DLL_RESULT_OK no error occurred If i is positive SigMath calls the function char GetText_V int in the DLL if existing with argument i If GetText_V returns a non zero pointer the corresponding text is returned to the user Otherwise SigMath produces an error message containing the name of the function and DLL and the value of i If i is negative SigMath closes the current document The function GetText_V is not accessible by the user Suffix Return type of Integer _R Real _C Complex _B Bool _V Void S String _VP Pointer Tbl 8 1 Possible Suffices for DLL functions and properties 153 8 Using and Writing DLLs The arguments of the function have the following meaning argType describes the types of pointers contained in argPointer The argument argSize contains sizes of arrays As an example we consider a function which has two arguments The first argument is a real number argType 0 DLL_ARG_DOUBLE argType 1 argSize no entry 0 dimension
136. logously ctNameY determines the vertical coordinates The displayed axis ranges are determined by the values of the functions on the intervals first last with respect to X and Y resp In the fourth variant the lengths of both vectors have to match After introducing the basic plot commands we want to add a plot to an existing document As we already pointed out all plot commands for a two dimensional plot have to be put in a plot2d environment Before SigMath can display the plot we also have to specify a window which con tains the plot This is done using the properties PlotWidth for the width and PlotHeight for the height given in relative screen coordinates 1 stands for the whole screen width or height 0 5 for half the screen Minimum is 0 01 and maximum is 10 If we do not specify the position of the window explicitely it will appear at the same position as it is written in the source code Otherwise we can specify its position using the property PlotPos The top left corner of the frame is given in relative screen coordinates i e e g 0 0 for the top left corner or 0 25 0 5 for a fourth of screen width to the left and half the screen height down begin plot2d PlotWidth 0 5 PlotHeight 0 5 PlotPos 0 25 0 25 Plot end plot2d Example 4 1 Let us have a look at a simple example We want to draw the graph of the sine function on the interval 0 27 Code Sine txt function f x return
137. lor 0 0 0 2 0 4 begin plot2d PlotBgColor 0 0 0 2 0 4 PlotBgStyle FILL PlotHeight 1 0 PlotWidth anim Height anim Width PlotHeight PlotPos 1 anim Height anim Width 2 1 0 PlotAxes PlotLineWidth 3 Plot 0 2 Pi g 0 2 Pi end plot2d end document We have to adjust the width of the plot to make sure that the circle really looks like a circle 54 4 1 Two dimensional plots Otherwise the plot frame and hence the axis are scaled according to the screen dimensions Typ ically the screen is not square We use relative commands that use screen dimensions provided by the operating system Another interesting aspect is that the second variant looks smoother especially at the points where the two semicircles join see Figure 4 2 The reason is that in the first variant sample points are uniformly distributed on the x axis Thus the rightmost and leftmost neighbors are too far away to generate a properly smooth curve To reduce this effect one has to increase the number of sample points In the second variant 128 points are uniformly distributed along the whole curve which results in a smoothing of the displayed curve In addition to creating a plot of an object the Plot command also returns the index of the generated plot item This index can be stored in a variable and can be used later on to modify several properties of the plot We continue this explanations at a later point 4 1 2 Color s
138. lues are possible gl GL_S gl GL_T gl GL_R and g1 GL_Q Since we treat only two dimensional tex tures we only use the first two genFunction specifies the function that is used to generate the texture To use reflection mapping we have to use GL_TEXTURE_GEN_MODE and the parameter genParameter GL_SPHERE_MAP 148 7 7 Textures Remark There are several other complex mapping techniques which can be used if we choose other values for the arguments in gl TexGen If we want to generate texture coordinates we have to enable this procedure in addition to the general activation of texturing Syntax Xgl Enable gl TEXTURE_GEN_S Xgl Enable gl TEXTURE_GEN_T Finally one should not forget to deactivate the texturing by using gl Disable Example 7 22 ANIMATED SPHERE WITH TREES SIGMATH EXAMPLE The mirror effect is produced by using the background wallpaper as texture too This and many other examples can also be found in the Example section of SigMath 149 7 Computer graphics Subdirectory Filename Functions KoordUtility txt SetPointKoordFromZyl SetPointKoordFromSph SetPointKoordFromKart Array Brackets txt DrawBrackets Brackets BracketsPart Rectangles txt DrawArrayRect VLines txt VLineRO Arrows Arrow2D txt Arrow2D Arrow2DDouble txt Arrow2DDouble Arrow2DBezier txt Arrow2DBezier Arrow2DBezierDouble txt A
139. lustrates both approaches variable P GLPoint 1 2 3 variable i 1 ValueOf P i ValueOf P In the first variant we display the value of the ith coordinate only and interactive modification is possible i may only take values in 1 and 4 The second variant displays GLPoint P x P y P z on the screen not allowing any manipulation Methods For GLPoint variables arithmetical operators are defined For addition subtraction and com pound assignments the same rules as for arrays apply The multiplication operator computes the inner product of the two vectors though Note that the fourth component is ignored in this computations Let us give some examples for arithmetical operators with GLPoint variables variable P GLPoint 1 2 variable Q GLPoint 2 4 6 8 variable R GLPoint 3 5 7 9 variable S P Q variable T R S The type of the variable T is detected on initialization as GLPoint Since P y 0 and the fourth coordinate is ignored the value of S is given by S 1 2 2 4 0 6 1 3 6 6 1 100 6 3 Complex data types The multiplication returns the value of the inner product i e T 2x3 4 5 6 7 88 The comparison operator is defined as for arrays Two GLPoint variables are equal if they equal in all elements Type conversion do not apply here since elements of a GLPoint vector are always of data type Real The fourth component is not ignored In addition to the class array the c
140. modified using gl TexParameter Syntax gl TexParameter target parameter value By specifying gl TEXTURE_2D as target all parameters of the current texture are modified The current texture in the example is the texture with ID TextureNr The different parameters its possible values and the effects of those values are summarized in the following table These parameters can be used to control the way the texture mapping is performed Points on the object are pixels points in the texture are called texels texture elements The coordinates of the object are denoted by x and y whereas the corresponding coordinates of the texture are denoted by s and t parameter value gl TEXTURE_MAG_FILTER gl LINEAR gl NEAREST gl TEXTURE_MIN_FILTER gl LINEAR gl NEAREST gl NEAREST_MIPMAP_NEAREST gl LINEAR_MIPMAP_NEAREST gl NEAREST_MIPMAP_LINEAR gl LINEAR_MIPMAP_LINEAR gl TEXTURE_WRAP_S gl REPEAT gl CLAMP gl TEXTURE_WRAP_T gl REPEAT gl CLAMP gl TEXTURE_MAG_FILTER If the texture is smaller than the surface the texture must be enlarged There are more pixels than texels making a one to one mapping impossible This parameter controls the way the color of the pixel is calculated On gl LINEAR the color of the pixel is computed as a weighted average of four neighboring texels whereas on gl NEAREST we use the color of the nearest texel to the pixel gl TEXTURE_MIN_FILTER This parameter controls the way a p
141. multaneous initialization of the variable i e we have to assign an initial value to it Syntax variable NameOfVariable InitialValue 74 6 2 Data types Identifier Description See pp Integer integers 78 Real floating point numbers 78 Complex complex floating point numbers 78 Bool boolean values 78 Void empty data type 79 Pointer memory addresses 79 String string of characters 79 and 84 Arrays arrays 92 Container complex data structures 102 GLPoint 4 dimensional vector 99 Bitmap color values of an image 105 TextEditor input field with the functionality of a text editor 105 Tbl 6 3 Data types in SigMath The reserved word variable tells SigMath to interpret the following as a definition of a variable NameOfVariable is a name which can be chosen according to some rules by the programmer The InitialValue can take different shapes The simplest variant is to specify a constant Basically all SigMath expressions that return a value of fixed data type see Sec tion 6 2 4 are valid Initialization defines the data type of the variable Therefore SigMath allows great flexibility which of course has to be handled with care For example the ini tial value can be specified using a function whose returned data type is specified on execution only Let us consider some examples for initializations and their consequences for the data types variable r 5 0 r is
142. n expression i e on evaluation it returns a value of data type Bool If not compilation of the document fails and an error message is displayed The evaluation of complex expressions like A x y needs further consideration Some programming languages use lazy evaluation i e as soon as the first condition A evaluates to true the whole expression is set to true In contrast SigMath evaluates all parts of the expression This way all subexpressions are investigated which pre vents some undesired effects SigMath also provides the command IgnoreIf that allows to ignore code if a certain condition is satisfied Syntax IgnoreIf condition code endif If the evaluation of the condition returns true the whole code segment within IgnoreIf and endif is ignored Multiple branches The if else structure only allows boolean expressions as conditions If one wants to check more than one condition at the same time we suggest to use the switch case structure It 107 6 Programming performs a distinction of cases One argument is evaluated by the following case commands If one condition is satisfied the corresponding code segment is executed To exit the execution of the switch case structure one can put a break command at the end of a command block This terminates the execution of the whole switch case block Alternatively more than one case may be satisfied and executed In analogy to else there is an optional default
143. n in connection with comparisons Whereas 2 2 0 returns true for scalar values the IntegerArray 2 2 and the RealArray 2 0 2 0 are not equal Besides and no other comparison operators are available for arrays 94 6 3 Complex data types Logical comparisons and bitwise operators are not admissible The increment operators and do not apply to single array components For write access on an array element one has to put the corresponding element on the left hand side of an assignment In case of arithmetical arrays also compound assignments are admissible Writing NA 3 results in an array whose elements all equal 3 Properties of arrays Given an array A we can determine its overall size or the size of one dimension by using the following methods Syntax A sizeQ A size i The special form size i returns the number of elements in the ith dimension Note that the method size for arrays has a different behavior than the Size method in the class string For arrays it is read only To change the layout of an array one has to use the Resize method The dimensionality of an array can be obtained by the Dim method Syntax A Dim To detect if the array has size 0 one can either write A size 0 or use the empty property If A size 0 it returns true Syntax A empty Every array has a version number that is initialized by O on definition Every modification of the array increases the version of the array Using
144. n means multiplying the matrix with the points We will explain the behavior of these 4x4 matrices in Section 7 5 and give some examples for transformations of objects there 7 4 1 Scaling Scaling an object results in stretching or squeezing an object Syntax gl Scale sx sy sz sx sy and sz are of type Real and positive If one of the values is between 0 and 1 the object is squeezed if the value is bigger than one the object is stretched in this direction Scaling is implemented as the following matrix multiplication sx 0 0 0 Sx Xz 0 sy 0 0 y _ syxy 0 0 sz 0 z 7 SZ Xz 0 0 0 1 1 1 135 7 Computer graphics 7 4 2 Translation On translations we move an object along a straight line which is given by a vector One can imagine that the vector is attached to every point and then the points are moved along the vector Syntax gl Translate dx dy dz To move a point we just have to add the vector dx dy dz The corresponding matrix multiplication reads as follows 1 0 0 dx x dx 0 1 0 dy y y dy 0 0 1 dz Z 2 z dz 000 1 1 if 7 4 3 Rotation A rotation is given by a rotation axis and a rotation angle The rotation angle is given is degrees the rotation axis is specified by a vector where the direction determines the direction of rotation It is given implicitly by the right hand rule see p 126 Syntax gl Rotate phi x y z The rotation matrix depends on the rotation angle and the rotation axis
145. n of objects can also be regarded as a transformation of the camera and thus we speak of positioning the camera This positioning is performed using the modelview matrix On the level of implementation we have two choices to perform the positioning 1 We move all objects using transformations in front of the camera as in Example 7 15 Since all transformations have to be given explicitely we call this method explicit 2 The glu library contains a command to position the camera glu LookAt This can be used to move or rotate the camera Since camera position is fixed we implicitly move and rotate all objects This command involves several matrix multiplications Syntax glu LookAt eye x eye y eye z vc x vc y VC Z vup x vup y vup z The eye coordinates determine the position of the camera vc stands for viewcenter and defines the center of the scene The direction of view is pointing from the camera position to the center of the scene The view up vector vup determines which side is up A view up vector pointing in the direction of the negative x axis e g results in rotating the scene by 90 degrees around the positive z axis All arguments are of type Real 7 6 2 Projection All transformations in this subsection are performed using a projection matrix It is used to limit our considerations to a cuboid in three dimensional space This is done by determining the so called clip planes i e the six faces of the cuboid All obje
146. nd displayed since the document has been started Syntax anim FrameNr The property Time measures the time an animation is running in seconds Syntax anim Time If the user pauses an animation by pressing CTRL Space Time and FrameNr remain con stant till the animation is resumed This can be used to restart animations after such a break at the correct time position The refreshment rate of the screen can be adjusted by the property FramesPerSec Mind that the actual refreshment rate depends on the number and type of calculations necessary the overall system and the current system load Syntax anim FramesPerSec Finally the property Busywaiting controls SigMath s waiting behavior Syntax anim Busywaiting After finishing all steps to display the next frame SigMath either runs an nearly empty loop Busywaiting true or uses the Sleep function of the operating system until the frame is displayed 6 2 Data types SigMath distinguishes between two kinds of objects constants and variables Constants see Section 6 2 1 have a fixed value and no identifier They are sometimes also called literals Variables Section 6 2 2 act as a placeholder for such values Such an object can change its content in course of time Every such object has a specified type the data type Section 6 2 3 and thus inherits all properties of this class Every data type has a number of properties and functions to modify them For example numbers
147. nd document To get the position of the mouse cursor the following functions are available Syntax Nui MouseX Nui MouseY Nui MouseDeltaX Nui MouseDeltaY Here MouseX and MouseY return the current position of the mouse cursor in relative screen coordinates MouseDeltaX and MouseDeltaY can be used to determine the difference of the mouse position from the last frame to the current frame These commands are read only To 47 3 User interaction set the position of the mouse cursor we can use SetMouseXY Syntax ui SetMouseXY x y Here x and y are numbers between 0 and 1 that specify the relative screen coordinates The can be given by time dependent functions allowing the mouse to move independent of user action To determine whether a mouse button is pressed we can use the commands IsLeftButton Down and IsRightButtonDown Syntax ui IsLeftButtonDown ui IsRightButtonDown To customize the layout of the mouse cursor we have to adjust the value of the property Cursor to one of the values contained in Table 3 4 Syntax ui Cursor Value Description crCross cross hairs crDrag pointing hand crHandPoint pointing hand crHourGlass hourglass crNo circle crossed crSizeNESW diagonal double arrow crSizeNs vertical double arrow crSizeNWSE diagonal double arrow crSizeWE horizontal double arrow crUpArrow top arrow crArrow standard cursor Tbl 3 4 Mouse cursor 3 5
148. ndow height given in pixels Perspective projection A more realistic view On a perspective projection the size of objects decreases on increasing distance to the camera The generated sceneries are more realistic Syntax gl Frustum left right bottom top near far Xglu Perpective visionY ratio near far Determination of the visible area is performed using gl Frustum The arguments have the same meaning as in the parallel projection The function glu Perspective produces the same result as gl Frustum but uses different arguments visionY gives the angle of the top clip plane to the y axis and ratio determines the ratio of width to height near and far determine the front and back clip plane In contrast to the previous usage the values must be positive INSERT PERSPECTIVE PROJECTION Based on the type of projection and the determination of the visible cuboid OpenGL computes the clip coordinates Those are used in the last step to determine the window coordinates which are actually used to display the scenery on the screen 140 7 7 Textures 7 6 3 Viewport Basis for the viewport transformation are the normalized device coordinates In this step the scenery is adapted to technical limitations of the system color gradients have to be reduced to a finite color palette straight lines have to be displayed using pixels The whole projection step is performed by OpenGL automatically The viewport command defines a rectangular area
149. o OpenGL functions and are thus treated in Chapter 7 101 6 Programming 6 3 4 Container Container is a general expression for complex data structures which can be used to organize and store data elements There are different types of data storage strategies which are implemented as different types of containers Each strategy has its advantages and disadvantages The choice of the correct container depends on the later usage Although the implementation of arrays and containers in SigMath differs we use arrays to introduce the fundamental concepts connected with containers Arrays allow fast access to single data elements using the index operator If we do not know the exact position of a single element in an array though the data structure does not allow for efficient search strategies Here a sorted data structure increases efficiency Such data structures are e g realized by containers of type Set As a complex data structure the class cont provides methods for containers There exist methods for all types of containers and specialized methods only applicable to containers of a certain type Definition All implemented types of containers can be found in the following syntax to generate empty containers Syntax variable S Set variable MS Multiset variable M Map variable MM Multimap variable L ListQ To use container variables in functions or as function arguments or for operators we have to s
150. o the operating system Basically events allow to interact on keyboard strokes or mouse movement The following commands allow to connect the corresponding events to functions Syntax OnDocClose fctName OnPageChange fctName OnVarChange varName fctName OnWinMessage fctName The argument fctName contains the name of the function that is called on the occurrence of the corresponding event For example if the user wants to close a document the boolean func tion fctName is called if one uses OnDocClose Depending on the return value of the function the document closes or remains opened The command OnPageChange calls the function fctName which needs an argument of type Integer before switching to a new page The value of the argu ment is the page number of the target page As before the page is only changed if the function returns true After the user changes the value of the variable varName the function fctName is called if one uses OnVarChange In this case the value of the variable is already changed since the next frame is already shown The command OnWinMessage is the most flexible and most complex of the four commands Thus there are a lot of requirements on the function fctName As an example we consider the function ProcessMessage Syntax function ProcessMessage message argi arg2 commands A function in OnWinMessage needs three arguments and returns a boolean value If true is returned the message
151. oad array variables from or to files we use Save and Load Syntax NA Save filename NA Load filename Here either the content of A is written to the file with name filename or A is filled with the content of filename For the Save method the file name extension controls if the array is stored as an image jpg or bmp or as text With Load one can load an array which was previously stored by a Save command To display an array in closed form SigMath provides a method to convert arrays to strings Syntax A ToString s ToString stores in the variable s of type String a list of elements separated by blanks and or dered in the way they are stored For matrices the line break is depicted For higher dimensional arrays the structure is lost and the resulting string contains only one single line Within the document environment ArrayBlock is a comfortable alternative to ToString Syntax ArrayBlock name The argument name is a String that gives the name of the array The following lines of code display the array A on the screen according to its layout For example a 2 x 3 matrix is given in 2 lines and 3 columns variable A IntegerArray 2 3 begin array ArrayBlock A end array This type of display can only be used for one or two dimensional arrays i e vectors and matrices An array of higher dimension must be rearranged using the Resize method before displaying it with ArrayBlock For the converse tas
152. om left corner of the screen This multiplication can be interpreted as viewing the object through the lense of a camera Thus the modelview matrix is used to determine the position of the camera Further options to modify the position of the camera will be treated in Section 7 6 1 The result of the multiplication of the object coordinates by the modelview matrix are the so called eye coordinates The camera is placed at the origin of this coordinate system Using a projection matrix gl PROJECTION we generate the cuboid we want to look at It is then given in the clip coordinate system The transformation matrix between the eye and the clip coordinates is just given by the projection matrix More information on the technical details is 138 7 6 Scenery given in Section 7 6 2 On basis of the chosen perspective see Section 7 6 2 the normalized device coordinates are computed Here sizes and thus coordinates are determined according to the distance of the object to the observer Finally the window coordinates are the result of the viewpoint transformation This de termines the area of the screen which contains the scene we want to display The scene is automatically scaled to the size of the viewport Section 7 6 3 will give more details on this last step 7 6 1 Positioning of the camera Basically the position of the camera is fixed at the origin pointing in the direction of the negative z axis Nevertheless every transformatio
153. omparison operators lt lt gt and gt are defined for GLPoint variables elementwise Using the method Length computes the Euclidean length of a vector SquaredLength com putes is square Syntax P Length P SquaredLength Here length is the Euclidean length of the three dimensional vector obtained by ignoring the fourth coordinate The returned value has data type Real By this method the length of a GLPoint variable can be adjusted too If we assign a new length to a GLPoint variable P by P Length x the vector is rescaled preserving its direction to the new length x The second method SquaredLength only allows read access To determine the angle between two vectors we can use the methods AngleDegree and An gleRadian The first method returns the angle in degrees the second in arc length Syntax XP AngleDegree q XP AngleRadian q Both methods need an argument q of type GLPoint and do not change P If one of the vectors is the zero vector both methods return 0 The method CrossProd allows to save the vector cross product of two vectors to the calling variable Syntax P CrossProd a b Here a and b are two GLPoint variables The vector P is then orthogonal to a and b and its length is determined by the area of the parallelogram which is spanned by a and b As before the fourth coordinate is ignored in the computations and in the resulting vector P w equals 1 The remaining methods serve as interfaces t
154. on Clicking e g the link Text1 the variable Text1 appears after the colon Clicking Text2 afterwards the displayed text changes to Text2 accordingly 3 2 Input fields Input fields provide a comfortable way to enter numbers or change them gradually In particu lar SigMath allows to modify the value of a parameter by pressing the right mouse button and moving the mouse up or down Input fields are generated by using the Input command Syntax Input nameOfVariable The value that is entered or changed by the user is assigned to the variable nameOfVariable To this end the variable has to be defined and initialized beforehand We can specify the way of input e g determine the number of decimal places displayed or if there are limitations on the value of the variable or step sizes The number of decimal places is determined by the Digits property Syntax Digits Here 1 stands for one decimal place 3 would be three decimal places and a negative value makes the number disappear The acceptable ranges of values of nameOfVariable are determined by the three properties InputMin InputMax and InputDelta Syntax InputMin InputMax InputDelta InputMin and InputMax can attain arbitrary real numbers If the user tries to enter a number smaller than InputMin or larger than InputMax the displayed input variable is automatically set to the appropriate bound Mind all variables have to be known to SigMath before using Input see
155. on and sizes its current layout The function RecallResize can change the value of the variable dataP The function Message_V must be contained in the DLL It allows to react on SigMath mes 154 8 2 Writing DLLs sages and is only used internally A possible implementation is given in the following __declspec dllexport int Message_V int type void argPointer switch type E case DLL_MSG_AFTER_LOAD Resize RECALL_RESIZE argPointer 0 further initializations break case DLL_MSG_BEFORE_DELETE clean up break return DLL_RESULT_OK The first argument of the function is the message type the value of argPointer depends on the type Currently only two types are implemented The message DLL_MSG_AFTER_LOAD is sent after loading the library argPointer contains pointers to the following SigMath functions argPointer 0 Resize argPointer 1 GetFunction argPointer 2 FreeFunction Instead of writing 0 1 2 one can also use the following symbols defined in ZS_ArgTypesDLL h DLL_INDEX_RECALL_RESIZE DLL_INDEX_GET_FUNCTION and DLL_INDEX_FREE_FUNCTION 2 The message DLL_MSG_BEFORE_DELETE is sent before removing the library here argPointer 0 Example 8 3 Let us have a look at an example that changes the layout of an array The following function writes the content of the file FileName to the matrix D __declspec dllexport int LoadArray_V int argTypel int argSizel void argPointer
156. ons their value is stored temporarily during function evaluation and is no longer accessible after the execution 6 6 Multithreading In order optimize code one should think about certain subtasks than can be run simultane ously concurrently The class thread provides some commands to define and run independent threads We can either execute different functions or several instances of one function in concur rent threads In order to associate a function to a thread we write thread BeginThread Syntax thread BeginThread functionName Here functionName gives the name of a function without arguments The associated thread stops as soon as the execution of the function is terminated All threads are registered within 114 6 6 Multithreading the document On closing of the document all threads are terminated and an error message is returned This may result in inconsistencies Thus we recommend to make sure that all threads are terminated in a regular way Example 6 9 In the following example two threads are active SigMath itself and the thread associated with the function Count variable Stop false variable i 0 function Count while Stop i thread BeginThread Count function CheckThread if Stop return true MessageBox0K false Click Stop to finish all threads return false begin document OnDocClose CheckThread i ValueDf i Stop Input Stop end
157. pecify the data type of a container see Section 6 5 Syntax variable cont S Set variable cont MS Multiset variable cont M Map variable cont MM Multimap variable cont L ListQ We give a list of the implemented container types and a short description in Table 6 11 All introduced container classes share the following properties 1 Every container may consist of an arbitrary number of data elements In contrast to arrays we do not have to specify neither size nor data type beforehand This guarantees a high level of flexibility and storage efficiency The only limitation on mixing data types is that 102 6 3 Complex data types Type Description Set A Set contains a sorted set of data elements Every element must be unique This restriction increases the complexity for inserting new ele ments On the other hand fast access to data elements is provided MultiSet In addition to Set the data structure Multiset allows elements to appear multiple times Map A Map contains elements consisting of its value and of a key The elements are sorted with respect to their key Every key must be unique This way of storing data allows fast access of the elements of the container MultiMap In addition to Map the data structure MultiMap allows keys to appear multiple times List A List stores its elements unsorted but in a determined order This data structure is optimized for fa
158. r rows automatically To ask for the value of certain elements of the array we need to have access to a certain array This is done by referencing to the index of an array We suggest to save it to a variable of type Integer within the corresponding array environment 36 Xbeginfarray entries of the array arrayInd ArrayIndex end array 2 10 Symbols If we want to detect whether there is an entry of the array enviroment with index arrayInd at a point having pixel coordinates x y we use the ArrayGetCell command It returns true if the condition is fulfilled and writes column and row index to the variables row and col If there is no entry at this point row and col remain unchanged and ArrayGetCell returns false If there is no array environment with index arrayInd at all an error message is returned Since the draw functions in the example are OpenGL commands we postpone their description to the corresponding Chapter 7 2 10 Symbols Known to ATFX users SigMath provides several commands to insert frequently used symbols in the text They are given in the following Table 2 5 Greek lower case letters a alpha A Nlambda T tau B beta p mu v upsilon y gamma v nu o phi delta E xi x chi e epsilon O No w psi zeta m pi w omeg
159. ray cont glpoint and string that implement complex data types see Section 6 1 and Table 6 1 Example 6 8 Let us give examples for function definitions The first function f has two arguments c and A and returns the value of the expression c 2 A The argument A may be a primitive data type or an array All commands within the function body can handle both cases 113 6 Programming Sum array A is built to sum up all elements of an array Since the function body uses the method A size and the index operator A i we specify in the function header the class array Code complexArguments txt function f c A return c 2 A function Sum array A variable Result 0 for variable i 0 i lt A size i Result A i return Result For arguments in a function call SigMath uses call by reference That means that the values of the arguments are not stored in the memory separately Thus changing the value of an argument in the function body changes the value of the variable used in the function call The following example shows the effect of the expression f x on the value of x although x does not appear on the left hand side of an assignment function f a a 10 begin document variable x 4 f x Value0f x end document Since write access on arguments may affect global variables one should be careful about that If the function call involves constants or other expressi
160. re two types of primitives for quadragons g1 QUADS This is the simple type Similar to gl TRIANGLES every four consecutive vertices are connected to form a quadragon Example 7 9 Ngl Begin gl QUADS gl Vertex 100 gl Vertex 200 gl Vertex 200 gl Vertex 100 gl Vertex 100 gl Vertex 200 gl Vertex 200 gl Vertex 100 gl End This example produced two quadragons 1 2 3 4 and 5 6 7 8 where the vertices 3 and 6 and 4 and 5 coincide One environment can be used to define more than one object g1 QUAD_STRIP In this type two consecutive quadragons glue together The first four vertices build the first quadragon Every following quadragon uses the last two vertices of the preceding one in reverse order and the following two vertices of the list Example 7 10 gl Begin gl QUAD_STRIP gl gl gl gl gl gl 128 Vertex 100 Vertex 200 Vertex 200 Vertex 100 Vertex 200 Vertex 100 100 100 200 200 200 200 300 300 100 100 200 200 300 300 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 2 Primitive gl End This definition generates two quadragons 1 2 3 4 and 4 3 5 6 resulting in the same geomet rical object as in Example 7 9 We save two vertex definitions This type allows to define one object per environment only Polygon A polygon is a surface having arbitrary many vertices As before we have to
161. reen width in width and half of the screen height begin document ImageStretch FILL Image Path Image jpg 0 2 0 5 0 end document Example 2 3 To place the figure at a certain position in the rectangle we write the fol lowing begin document ImageStretch PROP Image Path Image jpg 0 1 0 2 0 5 0 end document Example 2 4 Finally the rectangle is the whole screen If the original figure is small enlarging it by a large factor usually produces poor quality and graining Thus in this caes we bound the maximal scaling to 2 begin document ImageStretch PROP_MAX_2 Image Path Image jpg 0 1 1 1 0 end document 31 2 Word processing 2 8 Colors and background For presentations on screen color is a fundamental tool Thus we explain the general principles of color usage in SigMath at this point SigMath uses the RGB color model Therby RGB stands for the colors red green and blue The RGB model is an additive color model where the three basic colors add to white optical mixing A color is specified by three numbers its red green and blue component Every component varies between 0 and 1 Thus all commands concerning colors have this three color components in the prescribed order as arguments For example 1 1 1 codes white and 0 0 0 codes black 1 0 0 returns red 0 1 0 green and 0 0 1 blue Variation of the color components yields all colors which can be displayed Addit
162. responding output format Now as usual includegraphics adds the figure to the ATEX document Another alternative is to generate the corresponding ATX code directly This con tains the text of the figure and the import of the corresponding EPS file Be careful on hiding text inserting axis labels at the right position might be a tedious thing to do Thus spending some time on the decision which application to use to insert text may save some work Most of the time exporting the whole figure to EPS and include it to TFX in the described way will generate acceptable results Font This option allows to select the font family for the text in the figure This can also be done in the source code by issuing the g12ps FontName command If positioning of text plays a role we recommend using the monospace font Courier This increases the chance of matching font metrics which are used to calculate the positions of text blocks Sorting Using this option one can select an algorithm to detect the position of an object with respect to the observer and hence which objects cover other objects Suitable for most 2D graphics is the option NONE for most 3D figures Simple will produce acceptable results The most accurate but computationally most extensive algorithm can be selected by BSP We suggest to start with the low level settings and increase complexity until the resulting figure matches the requirements 67 5 Exporting graphics
163. resulting plot drastically The syntax looks as follows Syntax Plot fctName first last xmin xmax Plot vectorName xmin xmax Plot fctNameX firstX lastX fctNameY firstY lastyY XPlot vectorNameX vectorNameY The first variant computes the graph of a previously defined function fctName on the in terval first last If no further arguments are given the function is drawn on that inter val Otherwise it is only drawn on xmin xmax The visualization uses equidistant sample points of the function in the interval The number of points can be changed using the property PlotNumPointsX2D Basically in all variants the points are connected by lines piecewise linear interpolation Thus if PlotNumPointsX2D is low even smooth functions may look non smooth In the second variant we pass the name of a vector vectorName as the first argument Its entries are drawn versus the discrete x axis starting with 1 up to the length of the vector If 5l 4 Plots in SigMath optional arguments xmin and xmax are specified the points are only drawn on the interval xmin xmax The analogy to the first variant is obvious since evaluating a function at finitely many points just generates a vector containing the function values The third and fourth variant use arguments in parameter representation Here fctNameX denotes a function that is evaluated on the interval firstX lastX and determines the hori zontal coordinate of the image Ana
164. returns the value of the first element of the array the back method returns its last value Here first and last refer to the way of storing arrays in memory They are stored sequentially in a connected block in memory and the interpretation of multiple dimensions i provided by using index tuples The order of storage is given by running the index of the last dimension first and so on We illustrate the principle on a 2 x 2 x 2 array A aooo 4001 Goo 011 Q100 101 110 111 For example a two dimensional matrix is stored by concatenating its rows This way A front O returns the value of ago and A back the value of a111 Operators on arrays Arithmetical operators see Section 6 2 4 are also defined for arrays Besides data type compat ibility the layouts of the operands have to match That means two arrays can only be added or subtracted if the number of elements equal The resulting array has the same dimensions as the first summand The operator results in a matrix multiplication and is thus available for two dimensional arrays with the following layout rxn nxd This command results in a rxd matrix Automatic type conversion is performed as for elementary data types see Table 6 7 Taking powers or division of arrays is not implemented but scalar multiplication is performed elementwise Two arrays are equal if they have the same dimension size type and match in all elements This definition excludes type conversio
165. riginal view Fig 3 1 Four squares and mouse motion b after motion 4 Plots in SigMath This chapter deals with the visualization of mathematical objects like functions vectors We will learn to define axis and how to modify their properties add or change background and how to modify the resulting graphics interactively With this simple techniques we will be able to generate highly detailed graphics In order to tell SigMath s interpreter that the following commands are meant to generate a plot we enclose that code in plot2d and plot3d environments respectively A plot2d environment generates a plot which has an identifier a number for later reference Within a plot i e within a plot2d environment we draw graphics using the Plot command Every Plot command adds a plot item which again gets a number as identifier Thus every single plot item within a plot can be modified later on without having to redesign the whole plot again 4 1 Two dimensional plots 4 1 1 Generating plots We begin with the generation and modification of two dimensional plots Basically all we need to generate a plot is the Plot command Depending on its calling arguments SigMath auto matically chooses the appropriate way to visualize the object Thus Plot applies to functions and vectors in canonical or parameter representation Every plot item generated from a Plot command has a number of properties that can be modified to change the look of the
166. rom a given String we can use the method SubString Syntax S Substring index count Here SubString returns the String starting at index index of S The optional argument count specifies the length of the substring If index exceeds the length of S or count is non positive then the empty String is returned The SubstringBefore method returns the substring in S that stands before the first appear ance of the specified text Syntax S SubstringBefore s Here s contains the separating text e g dot that separates the filename from its file name extension The SubstringBefore method only works as a function and leaves S unchanged and the returning value has to be stored to another variable If S does not contain s as a substring the method returns the value of S The methods Position and PositionCI allow to search for the starting index of the first appearance of a substring 86 6 3 Complex data types Syntax S Position index s S PositionCI index s Here s specifies the text to be searched for and the optional argument index allows to specify an index where the search starts in the String As before a negative index is automatically mapped to 0 and if index is larger than S Size an error message is returned The first variant Position is case sensitive the second PositionCI is case insensitive To insert a new substring in an existing string we can use Insert Syntax S Insert index s Here index has to
167. rrow2DBezierDouble Arrow3D txt Arrow3DO Line txt Line O TubeClosed txt TubeClosed Bodies Cube txt DrawCube CubeTexture txt TexturedCube CubeTheColor txt TheColorCube Gear txt Gear Parallelepiped txt ParPiped ParallelepipedTubes txt ParPipTubes Circle Circle txt Circle2D PartialCircleAA2D txt PartialCircleAA2D PartialCircleArrow2D txt PartialCircleArrow2D PartialCirclePA2D txt PartialCirclePA2D Ellipse Ellipse txt Ellipse2D Figures Ikosaeder txt Ikosaeder RegularPolygon txt RegularPolygon RegularPolygonRect txt RegularPolygonRect Lines GridLines txt GridLines LineBezier2D txt LineBezier2D 150 Tbl 7 2 provided functions 8 Using and Writing DLLs Import of external libraries so called dynamic link libraries DLL for short can increase the functionality of SigMath We will give a short description of how to import existing DLLs or how to generate DLLs for this purpose 8 1 Loading DLLs To make functions or constants contained in an external DLL available within SigMath we use the command syst LoadLibrary Syntax syst LoadLibrary filename 1libName This command loads the external library filename Since DLLs can contain malware the user is asked to accept this step Commands and constants of the DLL may then be used by libName command The second argument libName has to be a valid class name and must be different from already used class or library identifiers
168. rs The part in the brackets after is called the function body All commands in the function body are executed on the function is called until we reach the first return command Thus return has two effects the variable or constant following it is the returning value of the function and return closes the function All commands after the first return will not be executed Nev ertheless a function might contain more than one return command e g within if or switch commands Since the data type of the return value need not be specified functions provide a high level of flexibility Recursive function calls are not yet implemented in SigMath Thus a function cannot call itself This does not exclude expressions of the form f f x since this is in fact an iteration The return value of the inner function call is the input value of the outer call as long as the return value matches the specifications of the arguments in the definition of f Function arguments Let us make some comments on the usage of function arguments Typically the data type of an argument need not be specified Thus arguments having the same name but different data types may appear If one wants to use certain methods for complex data types on the arguments in the function body it is advisable to specify the data type of the argument in quotation marks in front of the identifier in the function header In particular we recommend this method for the classes ar
169. sa ed o a wed Te Rie TEXTES to ps e o Ere ae de tes AR 7 7 1 Basic principles o 7 7 2 Types Of textures e 7 7 3 Loading textures e 7 7 4 Mipmapping e 7 7 9 Texture mapping 8 Using and Writing DLLs 31 Loading DEES tad adi 8 22 Writing DES iii a ad oe 9 Rendering Index Contents List of Figures 1 1 1 2 2 1 2 2 2 3 2 4 3 1 4 1 4 2 4 3 4 4 4 5 5 1 7 1 Starting up SigMath 2 ee 14 FiditOr WindOW 32 205 a adem Bia Bee RS eo AE ee a 15 The page layout template Standard with date multipage 22 Effect of the page format parameters 2 o 24 Positioning of text using TextPos aoaaa aa o 29 Example for a color gradient o oo a 34 Four squares and mouse motion 2 o a 50 The sine function on the interval 0 27 225 2 06244 200445054425 53 Plotting a circle in two different ways 1 0 0 00 e 54 PTOtStyle Options agan elisa eo as te Aya eid de de ler ee a ae A G 56 Redrawing plots depending on user interaction 2 02000 59 Placing tick marks manually oa aaa 62 The graphics export GUI e 66 Lines with different line patterns ooa oaa ee 131 List of Tables 1 1 Men items inthe editor vera Se a ta A 16 1 2 Shortcuts in the edit ccsa s raceri a aca ara ire Ga a a a eS 17 1 3 Commands that return file locations 2
170. se of binary operators the operator is placed in between the two arguments One simple example for an operator for Integers is the addition operator Arguments of an operator are typically called operands An expression is a concatenation of variables constants and function calls which evaluates to a certain value Every such part is an expression on its own and combinations of expressions are expressions All expressions determining a data type on evaluation can be used to initialize variables see Section 6 2 2 In particular that includes function calls or expressions connected with operators can be used for that Before introducing the available operators in SigMath we fix some terminology Depending on the number of operands we distinguish unary and binary operators Unary operators have one operand only e g sign operator while binary operators such as the addition need two operands Every operator has a priority which allows to define an order of execution within one expression The highest priority is 1 As is known from elementary calculus the multiplication operator has a higher priority than the addition operator Basically there are two ways to manipulate the order of execution determined by the pri orities of the operators The first way is to enclose subexpressions by parenthesis Then the operands in the expression in parenthesis are executed first where their execution order within the subexpression is determine
171. sed to store the RGB color coordinates The function LoadPower2 automatically fills the corresponding values from the file Working with procedural textures we fill the array manually by specifying for every single pixel its red green and blue color coordinates We can access the single elements with index triples as usual 145 7 Computer graphics for arrays Mind that the first row of the array index 0 is the bottom line of the image in display Finally h and w should be powers of 2 such that we can use TexImage2D without having to rescale the image beforehand Example 7 18 The following code generates a chess board Code Chess txt variable Image Bitmap codef Image Resize 32 32 for variable i 0 i lt 16 i for variable j 0 j lt 16 j Image i j 0 49 Image i j 1 49 Image i j 2 49 for variable i 16 i lt 32 i for variable j 16 j lt 32 j Image i j 0 49 Image i j 1 49 Image i j 2 49 for variable i 16 i lt 32 i for variable j 0 j lt 16 j Image i j 0 250 Image i j 1 for variable i 0 i lt 16 i for variable j 16 j lt 32 j Image i j 0 250 Image i j 1 250 Image i j 2 250 250 Image i j 2 250 The Bitmap variable Image can be used in the same way as theBitmap variable we filled using the function LoadTexture with the help of LoadPower2 Using text as textures SigMath provides a very comfortable way to convert text into texture Th
172. sin x begin document BgColor 0 0 0 2 0 4 Color 1 0 1 0 1 0 1 0 begin plot2d PlotWidth 0 8 PlotHeight 0 8 XPlotPos 0 1 0 9 Plot 0 2 Pi 52 4 1 Two dimensional plots end plot2d end document The resulting SigMath output is depicted in Figure 4 1 Fig 4 1 The sine function on the interval 0 27 Example 4 2 The second example gives to alternative ways to display a circle The first variant builds the circle from two semi circles described by the functions y 9 f x vV1 2 Code 2Semicircles txt function f x return sqrt 1 x x function g x return sqrt 1 x x begin document BgColor 0 0 0 2 0 4 begin plot2d XPlotBgColor 0 0 0 2 0 4 PlotBgStyle FILL PlotHeight 1 0 PlotWidth anim Height anim Width PlotHeight XPlotPos 1 anim Height anim Width 2 1 0 PlotAxes 1 53 4 Plots in SigMath a Two semi circles Mind kinks at the left and b One circle in parameter representation at the right Fig 4 2 Plotting a circle in two different ways AutoTicks PlotLineWidth 3 Plot 1 1 Plot g 1 1 end plot2d end document Next we want to draw the circle using its canonical parameter representation As we know the points on a circle fulfill the sine cosine law Code ParaKreis txt function f x return sin x function g x return cos x begin document BgCo
173. space fonts for such plots More information can be found in the Chapter 5 on p 67 4 1 7 Summary Table 4 1 contains all properties we introduced in this section in connection with two dimensional plots It serves as a short reminder More information is contained in the online help of SigMath 4 2 Three dimensional plots 63 4 Plots in SigMath Property Admissible values Default value PlotWidth between 0 and 1 1 0 PlotHeight between 0 and 1 1 0 PlotLineWidth positive integer 1 PlotLineStipple factor and String of length 16 consisting of 0 and 1 undefined PlotPointSize positive integer 1 PlotMargins array of 4 entries each entry between 0 and 1 0 1 0 1 0 1 0 1 TickLengthFactor arbitrary number 1 0 PlotNumPointsX2D positive integer 128 PlotNumPointsX3D between 0 and 1 32 PlotNumPointsY3D between 0 and 1 32 Interactive true or false false PlotAxes String consisting of at most 3 digits 123 AutoTicks String consisting of at most 3 digits 123 PlotStyle LINE_STRIP FILL RECT or FILLED_RECT LINE_STRIP PlotBgStyle NONE RECT or FILL NONE PlotColor color in RGB representation white 1 1 1 PlotBgColor color in RGB representation black 0 0 0 AxesColor color in RGB representation white 1 1 1 AxesDepth between 1000 and 1000 0 0 PlotDepth between 1000 and 1000 0 0 64 Tbl 4 1 Plot properties 5 Exporting graphics
174. st insertion and deletion at arbitrary locations Tbl 6 11 Container types in SigMath they have to be comparable That means the comparison operators and lt must be applicable for any two elements of a container 2 If we insert a data element A into a container only its value is inserted Thus altering the value of A does not affect the content of the container 3 The elements of a container are ordered The way they are sorted depends on the type of container The type Set guarantees an ascending list elements of List are appended to the end of the list 4 In contrast to arrays operators on containers are not secure That is the programmer of the code has to make sure that all operands of the operators lie within admissible ranges Access to elements in a container is only possible by so called iterators Those methods guar antee that the data structure is always in a well defined state Internally an iterator is a pointer which addresses the content of a container depending on its type Iterators allow to navigate to certain positions within the container and allow to read data or insert new elements The iterator represents a position within the container and can be used as an argument for a method Let us have a look on an example Example 6 1 We define the container variable S The container Set has the property that its elements are stored in sorted order Every time we include elements by insert to S t
175. st listName listMode gl commands gl EndList The variable listName is of type Integer and gl GenList assigns a uniquely determined identifier to it The argument numList is also of type Integer Simply writing variable listName 1 is also possible but the usage of gl GenList guarantees that every identifier is unique and thus prevents lists from being overwritten gl NewList opens the definition of the list The argument listMode can take the values gl COMPILE and gl COMPILE_AND_EXECUTE This results in either just compiling the list of compiling the list and executing the result directly 7 3 2 Calling lists To call a list one uses the following commands 134 7 4 Transformations Syntax gl CallList listName gl CallLists listNameArray One can either give one Integer listName or a IntegerArray listNameArray to call one or multiple lists respectively To prevent runtime errors one can check the existence of a list with gl IsList Syntax gl IsList listName This function returns true if the list exists and false otherwise Lists may contain lists This allows to generate complex sceneries from sub sceneries which are already given as lists Example 7 14 HERE AN EXAMPLE FOR A LIST 7 4 Transformations Transformations include scaling translation and rotation of points A transformation changes the so called active OpenGL matrix which is applied to all following objects before they are drawn Applicatio
176. subdirectory Shared OpenGL Array of the program directory In any case the following line has to be added in the preamble to introduce the Brackets command to SigMath Execute ProgDir Shared OpenGL Array Brackets txt Then Brackets is available Just place it at the end of the array environment that contains the array Syntax Brackets Changing the entries of matrices and vectors on the screen Using the InputArrayBlock command we can combine an array variable with the screen display in such a way that the user can change the value of the variable by direct input More on this topic can be found in Section 3 3 35 2 Word processing Determining the position of a cell SigMath does not provide further commands to modify the layout of a table directly However experienced programmers may use OpenGL for this purpose One typical task is to put a frame around certain rows or columns in a table In order to specify all necessary lines one has to determine the position of the text cells For this purpose SigMath provides the commands ArrayRect ArrayColX and ArrayRowY The position of the currently selected entry is given by ArrayGetCell The last command requires the index of the array which is obtained by ArrayIndex analogous to PlotIndex see Section 4 1 3 Syntax ArrayRect coord ArrayRect row1 col1 row2 col2 leftX leftY rightX righty ArrayColX coord ArrayRowY coord ArrayIndex ArrayGetCell x
177. t gl OpenGL glpoint 3D and 4D Points glu OpenGL utility library GLU gluti OpenGL help ui Graphical user interface re Renderer string String syst System thread Threads Tbl 6 1 Classes in SigMath random numbers Finally we distinguish between two types of commands that are related to certain class proper ties If the command can only be used to request the current value of the property it is read only Accordingly a command is write only if we can only assign a new value to a property and the command has the read write property if both actions are possible A comprehensive list of all methods and properties of the various classes in SigMath is given in the online help The anim class The class anim contains commands connected with animations Since they are fundamental but easy to understand we introduce them here For screen design there are five properties available Height Width Ratio and Screen Height and ScreenWidth Syntax Nanim Height Nanim Width 72 6 2 Data types Nanim Ratio anim ScreenWidth anim ScreenHeight Those properties give the height and the width of the SigMath window in pixels The third property gives the ratio of the first two properties The last two properties give the height and width of the screen in pixels Apart from that two further properties are connected with SigMath s animation history First FrameNr counts the number of frames SigMath computed a
178. tax Whereas LTRX aims at producing printed and therefore static documents SigMath s layout capabilities are designed for dynamic text display on the screen Additionally graphic export is available based on the GL2PS library by C Geuziane For developers SigMath provides a highly developed programming language which not only contains the usual control structures and complex data types but also allows the usage of OpenGL commands WINDOWS system routines and multithreading SigMath s functionality is easily ex tendable by importing functions and methods written in C C Summing up SigMath allows to generate animations performs the necessary mathematical computations displays text and enables the user to interact and navigate through the animation Although there might be applications designed for every single task we think that the mixture provided by SigMath makes it a powerful tool 11 1 Introduction 1 2 Basic principles All applications projects or presentations in SigMath consist of documents where a document is a plain text file in ASCIT format These files contain source code which tells SigMath what to show and how it should be presented If we open a SigMath document in SigMath the source code is automatically interpreted and the result is shown on the screen If the code only contains text it is displayed on the screen as it is In order to present formatted text additional commands have to be included in the
179. te 68 Option Description in case of state on occlusion cull all hidden polygons are removed from the target file reduces file size landscape the target file is in landscape instead of portrait no text all text is hidden see the remarks concerning ATFX export draw background the background is added tight bounding box the bounding box is put as close as possible around all ex ported objects simple line offset to all lines a small offset is added in z coordinate anti aliasing no pixmap all pixmaps are removed from the target file best root the construction of the BSP tree is optimized by using the root primitive yielding minimal number of branches no ps3 shading only on POSTSCRIPT disables the usage of shfill PosTSCRIPT level 3 operator that enhances the visual ization of smooth shaded primitives but might complicate later PDF conversion no blending opaqueness is disabled disregarding any commands in source code smooth shading only on POSTSCRIPT or SVG controls the number of flat shaded triangles that are used to approximate smooth shaded triangles needs three arguments nr ng nb which specify how many values with respect to the red green and blue color components are used in the interpolation a tri angle is subdivided recursively until the color difference in the color components on its corners is less than 1 nr 1 ng and 1 nb respectively Tbl
180. tells SigMaths interpreter to ignore the rest of the line For commands in the command regions we use the C C comment symbol e everything written in between and is ignored e and turns the rest of the line into a comment This distinction can be used to hide a lot of consecutive lines in the text region from the interpreter by commenting them We add a code section and put at the beginning and at the end of the section code text that should be hidden as a comment 1 6 3 Inserting documents If documents contain a lot of source code the contents tends to become confusing Thus one should split the document into several smaller portions and connect them using the Include command Thus way also allows to create and save templates for certain tasks that can be used over and over again The text file is included at the position of the Include command in the document and only executed on compilation Syntax Include filename The Include command requires the name of a text file filename as its argument In order to make this work one has to specify the correct path Table 1 3 contains a number of commands that make the task easier If the file to be included is contained in the same directory as the current document we write Include DocDir filename txt If we want to include the file Include txt which is located in a folder IncludeFiles in the directory Examples we have to specify the p
181. tes to true the following block of commands is executed otherwise SigMath looks for a command block beginning with else If now else block is specified SigMath contin ues with the execution of code following the if block Syntax if condition commands else commands In case that the if or else block consists of one command only the brackets may be omit ted Simple branches can be nested In that case every if block has to be accompanied by an associated else block The connection is determined by the structure of the nesting Let us consider an example Example 6 3 A simple if else structure can be used to display the reciprocal of a real number y We use if to compute the quotient of two numbers 1 Simple if else structure if y 0 0 result INF else result 1 y 2 Nested if else structure if y 0 0 if x 0 0 result NAN else 106 6 4 Control structures result INF else result x y 3 Computing the quotient without nested branches if y 0 0 amp amp x 0 result NAN if y 0 0 amp amp x 0 result INF if y 0 0 result x y Since the cases 0 0 and x 0 require special consideration we use nested branches in the second example This requires two comparisons in case that y 0 is true The last variant of computing the quotient has five comparisons since every condition has to be checked separately The condition must be a boolea
182. th for LeftMargin and TextWidth and screen height for TopMargin respectively Thus e g setting LeftMargin to 1 means that it equals screen width We indicate the distances that are controlled by the above properties by arrows in Figure 2 2 2 3 Fonts To change the font of text we use selectfont Further properties of the selected font are ad justed by the properties fontname fontpoints and fontcharset 23 2 Word processing opMargin The page layout is determinedby the commands This is the first line al LeftMargin TopMargin TextWidth MAA ooo orr They can be put everywherein the source code TextWidthis used only in compilation The value 1 of LeftMarginor TextWidthcorrespondsto screen width for TopMarginit correspondsto screen height Fig 2 2 Effect of the page format parameters Syntax fontname fontFamily fontpoints points fontcharset charSet selectfont The syntax corresponds to the commands in ATEX The argument fontFamily is the name of the font i e e g Arial for normal text Courier new for monospaced text Monospaced text is typewriter like i e every letter has the same width e g i and M Available font families depend on the operator system on your computer See the help pages of the system for more information The default font size in SigMath is 36 points For the displayed text this has to multiplied by the factor screen height in pixels 768 To change font si
183. the GoToString Syntax inputlabel here text and commands begin inputlabel here text and commands end inputlabel The first GoToPage or DoString command following this inputlabel environment uses the content of inputlabel instead of the value of text Afterwards the behavior of GoToPage returns to the default behavior By default every pages has a number by tracing link history and a name its label Number and label can be read by using CurrentPageNr and CurrentPageLabel respectively In this way we can link to a certain page Syntax CurrentPageNr CurrentPageLabel For example using the command line CurrentPageNr 2 the second page in the history of the current text document is displayed Be aware of the fact that this way of referencing pages is unreliable Thus we recommend the usage of labels 40 3 1 Hyperlinks Syntax PageLabel pageNr The command PageLabel returns the label of the page with number pageNr If no corre sponding page exists the empty String is returned Command links To combine functions with elements in the user interface we use the DoString command As with page links we can display colored text that executed a function if the user selects the text by left clicking Syntax DoString text fctName DoString text fctName ident The argument text is of type String and contains the text that should be displayed It is highlighted by using the color InputColor As w
184. the horizontal spacing to the next word by 3 18 of the height of the current font Command Description Factor Space Space NS 3 18 AS 4 18 AS 5 18 3 18 quad 1 qquad 2 Tbl 2 3 Shorthands for horizontal spacing 2 5 Positioning of text If we want to place text at a precise location on the screen we have to perform two steps First we have to specify the position of the text block and then display the text at this position 27 2 Word processing For the first task we can use the following commands Syntax XTextPos targetX targetY textX textY TextPos p textX textY TextPosPixel targetX targetY textX textY XTextPosPixel p textX textY The effect of those commands is as follows First a target point is specified that defines the origin for the positioning of the text block This is given the arguments targetX and targetY or by the first two entries of the floating point variable p see Chapter 7 The coordinates used in TextPos are relative screen coordinates i e 0 0 denotes the bottom left corner and 1 1 the top right corner of the screen TextPosPixel uses pixel coordinates i e 0 0 denotes the bot tom left corner and anim Width 1 anim Height 1 the top right corner Here anim Width and anim Height give the number of pixels in horizontal and vertical direction respectively see Section 6 1 The remaining two arguments textX and textY det
185. the primitives Primitives are e g vertices lines and polygons Using this basic items we can generate three dimensional sceneries The Section 7 4 introduces transformations and illustrates the way to create scenery The usage of textures i e bitmaps to decorate surfaces is then treated in Section 7 7 Finally we want to provide a small glimpse on the vast amount of methods to generate computer graphics provided by OpenGL 7 1 What is OpenGL OpenGL stands for Open Graphics Library and was developed by SILICON GRAPHICS It pro vides a platform independent and programming language independent interface to use the graph ics card The library only contains graphics operations In order to write code to generate computer graphics with OpenGL we have to make this interface accessible We do not have to take care of this link since SigMath is doing this for us All OpenGL commands can be identified by the prefix gl Other SigMath commands e g for word processing or programming do not belong to OpenGL They can be used as a framework to generate graphics At the moment the library provides about 250 commands to display real time animations of 3D sceneries It does not contain commands to generate complex three dimensional objects though All figures have to be built up from primitives In order to draw a circle e g one has to use a polygon SigMath provides a number of commands that allow to display certain common figures see Section 7 2 6
186. this initial texture as long as we do not bind it to another texture ourselves In this way we implemented a function to load textures The return value of the function is the ID of the texture Thus TextureLoad is mainly used in the definition and initialization of variables In our example roofTile is then the identifier of the loaded texture As indicated we introduce a second function called LoadTexture that provides basically the same functionality as TextureLoad In the implementation we use the OpenGL command Tex Image2D instead of the glu command glu Build2DMipMaps Syntax gl TexImage2D image Xgl TexImage2D filename There are two versions of this command image is a variable of type Bitmap and filename is the full qualified name path filename extension of the file containing the image This command does not rescale the given image and thus code has to be adjusted accordingly Template function LoadTexture filename w h w and h are redefined variable bmp Image Bitmap Image LoadPower2 filename w h 144 7 7 Textures variable TextureNr gl GenTextures gl BindTexture gl TEXTURE_2D TextureNr gl TexParameter gl TEXTURE_2D gl TEXTURE_MAG_FILTER gl LINEAR gl TexParameter gl TEXTURE_2D gl TEXTURE_MIN_FILTER gl LINEAR g1 TexImage2D Bild gl BindTexture gl TEXTURE_2D 0 return TextureNr variable w 0 0 variable h 0 0 variable roofTile LoadTexture
187. thus we can determine its normal vector We do not determine the normal vector for each surface separately but by defining the properties of the normal vector The rest is performed by the following glu function Syntax glu QuadricOrientation arg 133 7 Computer graphics This command sets the direction of the normal vector and thus specifies front and back side of the surfaces The argument takes the value glu OUTSIDE if the normal vectors should point to the outside Thus on glu OUTSIDE the front sides of the surfaces all point outwards Alternatively on glu INSIDE the normal vectors are pointed inwards The default setting is glu OUTSIDE Remark 1 For disks glu OUTSIDE results in a normal vector pointing in the direction of the positive z axis 2 The function glu QuadricNormals is important for the later rendering process We do not go into further details here 7 2 9 An extended example INSERT SCENE WITH PRIMITIVES AND ILLUSTRATE THE MEANING OF FRONT AND BACKSIDE 7 3 Working with lists With the help of lists one can group several OpenGL commands Besides improving readability of code it allows to display scenery efficiently In addition once we defined a scenery in a list we can draw the scenery by simply calling the list and do not have to rewrite the commands 7 3 1 Generating lists To generate a display list one has to write the following Syntax variable listName gl GenList numList gl NewLi
188. tion It returns the data type of its argument 76 6 2 Data types Syntax TypeOf argument As we will see later one does not have to specify the data type of an argument of a function in the definition Thus within the function body one can use TypeOf to determine the data type see Section 6 5 Accessing and displaying variables Since nearly all entries in a SigMath document are interpreted as text we have to use special commands to tell the interpreter to e g write out the value of a variable and not just its name or to assign a new value to a defined variable To access a variable x we simply put a backslash in front of the variable outside of code environments To display the value of a variable its internal format has to be converted to a displayable form e g numbers have to be converted from their internal number format to a sequence of digits or boolean values have to be rephrased as text All this conversions are performed by ValueOf Syntax ValueOf varName The argument varName is of type String that contains the name of the variable to be displayed As an example if we define the variable x and initialize it to 0 we display its value by writing the following line of code variable x 0 x WalueOf x By that a line containing the text x 0 is displayed on the screen since x and the equation sign are interpreted as simple text Scope and visibility of variables A special property of any v
189. tion the axis manually we can use the following commands The placement of tick marks can be specified using the PlotTick Syntax 60 4 1 Two dimensional plots XPlotTick axsNr position PlotTick axsNr first step numTicks The first variant puts a tick mark at position position on the axis with number axsNr and writes the corresponding value as text label The second variant puts tick marks and tick labels at the positions first k step where k runs from 0 to numTicks 1 The color of the tick marks is determined by the axis color the text label is given in the current text color The precision of the numbers in the tick labels can be adjusted by setting the Digits property see Section 3 2 Its integer value equals the number of displayed digits Setting Digits to negative values no text is displayed Besides the tick label we can adjust the way the tick mark looks Using TickLengthFactor we can scale the default length of a tick mark Using a negative value the tick is placed on the other side of the coordinate axis Example 4 4 To understand the effect of the previously introduced commands we want to have a look at the following source code The result produced by SigMath is depicted in Figure 4 5 Code PlotTick txt begin document begin plot2d AutoTicks AxisRange 1 0 10 AxisRange 2 4 10 PlotTick 2 4 1 15 PlotTick 1 2 Digits 1 PlotTick 1 4 Digits 1 TickLengthFactor PlotTick
190. tyle and background Let us introduce some of the properties a plot has All modification of these properties take only effect on the following Plot commands In order to change the color of a graph we assign to the PlotColor property the new color using its RGB coordinates see Section 2 8 The line width used to visualize the graph of a function can be adjusted using PlotLineWidth Syntax PlotLineWidth It can take any positive integer and is measured in pixels Furthermore the style of a graph can be modified To this end one can add one of the following lines to the source code Syntax PlotStyle LINE_STRIP PlotStyle FILL PlotStyle RECT PlotStyle FILLED_RECT PlotStyle POINT The option LINE_STRIP generates a line pattern using solid lines as default FILL fills between the graph and the x axis using grey color The option RECT has typically the most impact if one uses only a very low number of sample points The function is then approximated by step functions FILLED_RECT combines the effect of FILL and RECT i e it generates filled rectangles Finally POINT only draws the sample points without connecting them by straight lines To change the line pattern used in LINE_STRIP one can modify the PlotLineStipple property Syntax PlotLineStipple lineStippleFactor lineStipplePattern The pattern is given as a String of 0 and 1 s of length 16 Here 1 stands for a drawn pixel and 0 for an empty pixel Thus
191. wing plots depending on user interaction 59 4 Plots in SigMath 4 1 4 Axes Axes and its properties can be modified in a variety of different ways First PlotAxes defines which axis should be included in the plot Syntax PlotAxes PlotAxes 1 adds the x axis and PlotAxes 2 adds the y axis to the plot To draw both axes write PlotAxes 12 and for no axes at all PlotAxes As before the color of the axes can be changed using the property AxesColor Another im portant property is the range we want to display Without further specifying it the range is determined by the function or the vector we are going to show If we only want to draw a certain part or need a larger graphic to depict the function or vector we can manually define the range using AxisRange and AxesRanges The syntax is as follows Syntax AxisRange axsNr min max AxesRanges ranges The argument axsNr is Integer 1 or 2 the arguments min and max give the minimum and maximum of the corresponding axis interval The second variant takes a 2x2 array containing the bounds for the x and y axis The command GetAxesRanges saves the minimal and maximal values on the corresponding axis in an 2x2 array which is accessible by the specified variable identifier A Syntax GetAxesRanges A To place automatic tick marks and labels to the axes we use AutoTicks The syntax is anal ogous to PlotAxes Syntax AutoTicks If we want to parti
192. y consists of code that is executed depending on the loop header commands several times Counting loops Counting loops or for loops allow to repeat a certain code segment a fixed number of times Syntax for initialization condition reinitialization commands y A vital part of any for loop is the loop or counting variable It is initialized in the loop header i e before the loop body is executed the first time All commands in the initialization part are only executed once The condition typically contains the counting variable Depending on its truth value the loop body is executed In particular it may happened that the loop body is not executed at all After the last command in the loop body reinitialization is performed and the condition is checked with the new values of the variables Thus reinitialization is executed before entering the loop body a second time and then every time before reentering the loop As soon as the condition is not satisfied any more repetition stops and execution resumes at the next command after the the loop Example 6 6 Let us have a look at a very simple example The following code calculates the sum and product of the first four integers for variable i 1 i lt 5 i Sum i Product i The counting variable i is defined in the loop header and initialized by the value 1 Its scope is limited to the loop This memory saving way should be used any time the counting vari
193. ze assign the new font size measured in points to the property fontpoints Typically one is rather interested to rescale font size relative to the default size This can be done by using one of the commands listed in Table 2 2 They can be used in the source code directly All text following those commands then appears in the selected font size until font size is modified again or a new environment starts for environments see Setion 2 1 The property fontcharset can take the following values default ansi symbol chinese arabic greek and russian This property allows to load character sets used in the corre sponding language as long as the operator system on your computer supports them 24 2 4 Formatting Command Factor Sample tiny 0 33 tiny scriptsize 0 50 very small footnotesize 0 66 foot note small 0 75 small normalsize 1 00 normal large 1 33 large Large 1 50 larger LARGE 1 75 even larger huge 2 00 huge Huge 2 33 gigantic Tbl 2 2 Font scaling commands The new values of the properties fontname fontpoints and fontcharset only take action after the command selectfont All modifications apply to the text region only and only within the rest of the environment where they are specified Font shape can be changed by the following commands To produce bold face write bf for italics one uses it To underline or overline text use underline

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