User's Guide - Arizona Software
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1. or Prod The syntax is the same as for the sum Example 5 k 1 2 5 ara a k 1 2 7 5 or without equation editor k 1 5 n k k Integral or Integr The syntax is the following b J re tza or without equation editor Integr t a b f t Example 2 2 t wi fe erf x T 20 Derivative or 0 Syntax n times f or fon f vee Entering an Expression 13 Examples d f a f x Ox x a sin 1 sin z f03 x f x Note f g x F oxy a to So exp ax exp ax and not exp ax a Ox Ox Ox Iteration Syntax n times f n x f f f x Convolution Syntax f g Example f g x 0 201 dt Comments Syntax my comments Recognized Words Recognized Constants The following constants are recognized by Curvus Pro in MKSA units T 3 141592653589 e 2 718281828459 Euler number cLight 2 99792458 e8 speed of light in vacuum e0 8 85418782 e 12 permitivity of vacuum Entering an Expression 14 F 9 648456 e4 Faraday constant G 6 6732 e 11 Gravitational constant gamma 0 5772157 h 6 6262 e 34 Planck constant kBoltz 1 3807 e 23 Boltzmann constant Na 6 022 e23 Avogadro s number R 8 3144 gas constant U 1 6605655 e 27 atomic mass uo 4rne 7 permeability of vacuum For complex numbers i j x 1 And the logical con2. Curvus Pro version 3 1 User s Guide Ariz na September 2000 Arizona by Simon Bovet info curvuspro ch http Awww curvuspro ch Introduction 2 Introduction Presentation Curvus Pro has been developed in order to provide a quick user friendly and powerful tool for graphing any kind of two and three dimensional algebraic relation such as e Any explicit implicit or parametric curve and graph e 2D and 3D vector fields e 2D and 3D scalar density fields e Solutions of differential equations up to the third order e 2D logical relations The interface aims to be as transparent and easy to use as possible so that any user can rapidly and naturally take advantage of the many options provided by Curvus Pro The intuitive equation editor that offers an ideal way of entering mathematical expressions is just one out of the numerous original features included in Curvus Pro Also several visualization options are available such as e Several curves or graphs on the same document e Variable parameters for curve family plots corresponding to the different values of the parameter e Many vector drawing options etc Several numerical evaluation options allow calculating e Coordinates derivatives tangents osculating circles at any point of a curve a graph or a field e Roots extrema and inflexion point as well as intersections of 2D curves e Integrals over a finite or infinite range Numerous dynami
3. In this mode you can move the viewpoint location or navigate about or inside the graph Moving There are several ways of moving the viewpoint location e By clicking with the mouse at any direction in the document e By using the Number Pad keys e By using the right part of the toolbar above e By following a curve see paragraph Options below Dynamic Options 33 Note Keep the Shift Option Control or Command keys down to change the navigation speed Recording the Path You can record the a moving on by clicking on the icon e To pause the recording click on LLL Once a path is recorded you can follow it again by using the icons Lr a and You can also click on the progress bar to jump to the corresponding position Options I Click on the icon to make the following dialog appear O im atk and mri Teas Cio G tonie ined rameti Possions Gi aTa a eee ye ine e a _cemcet You can chose different playback modes loop back and forth and recording modes Transition e The Continue mode record each move The Positions mode record each position each time you click on the icon and then perform a smooth transition between each in the order they were recorded e The Follow the Curve mode only available for parametric curves and differential equation solutions defines a path follow the selected curve It cannot be modified Creating a QuickTime Movie Click on the icon to cr
4. Center Axis Root Find Integration Create Animation QuickTime Movie Z Axis Rotation Free Rotation Horizontal Offset Zoom In Evaluation Redraw Automatically Y Axis Rotation Grow Adjust Zoom Out Create Animation QuickTime movie Redraw Now Annexes 40 Parameters Toolbar Delete the parameters Definitions of the parameters Click on this icon to delete the selected parameters If no parameter is selected all parameters will be deleted e Tocreate anew parameter click ona empty line e Toselect a parameter click once on it Comment To select several parameters keep the Shift or Command key down Save the parameters Click on this icon to save the parameters in a file To modify a parameter double click on it Load parameters Click on this icon to load Move a parameter parameters from a file Click on on arrow to move a parameter in the list in order to have it sorted in the desired order Note See chapter Variable Parameters on page 18 for further information about using such parameters Annexes 41 Custom Definitions Toolbar 0 01 0 01 o B Delete the definitions Creation of the custom definition Click on this icon to delete the selected definitions If no definition is selected all definitions are deleted e Tocreate anew definition click ona empty line and type a new expression e T
5. by clicking on Rename or Erase Dimensions Click on the Dimensions tab of the Document Format dialog to modify the document dimensions Several predefined formats are available in the pop up menu Note You can also change directly the size of the document by clicking in the Size corner bottom right of the document window and keeping either the Option key or the Command key down Laying Out a Document 22 Adding Drawing Objects You can add several types of drawing objects texts circles rectangles lines marks equations etc to a Curvus Pro document in order to illustrate it by some drawings or explanations These objects can be selected moved and deleted as in any other common application with the mouse keeping the Shift key down for a multiple selection typing Backspace to deleted the selected objects Note You may use the Arrow keys to adjust the position of some objects Tools Toolbar You can use the Tools toolbar to add some vector drawing objects These are more or less the same as the one encountered in any other vector drawing application Click on one icon of the toolbar to select the corresponding object and click somewhere in the document window to insert it Note To keep an object selected in the toolbar double click on its icon The Object Menu Use the items Move Forward Move Backward Move to Front and Move to Back under the Object menu to modify the piling order of the sele
6. can enter the legend and the expression Expressiga Value Expression Z hi pression e meta Pried raian E Finde Radian Com Note The expression may contain any custom definitions Laying Out a Document 24 Modifying Axis and Grid Appearance in a 2D Document Modifying the Axis To modify the axis of the current document chose X Axis or Y Axis under the Format menu or just double click on one axis in the document window Following dialog appears then fel Wisible O begar behest TiciA irks 4 Lines HA Mpeg Ef Zero Every ho tuts 8 bliin Tice aa kek ef Lines I mering k perrh njor Tec kbar Click on the Range tab to change the range value the scale linear or logarithmic and the tick marks Click on the Format tab to change the appearance the position and the numbering of the axis Click on the Number tab to change the number format used for the numbering Select the Default item to use the default number format of the document see chapter The Preferences Dialog on page 48 Eventually click on the Font tab to change the appearance of the font used for the numbering Modifying the Grid To modify the appearance of the document grid chose Grid under the Format menu In the appearing dialog you can select the coordinates type rectangular or polar the main and second grid specifications as well as the background color and the document bounds Note To cha
7. exporting It is better to reduce the size of the original document before exporting the image dpi dots per inch Conclusion 51 The curve doesn t show up e The curve overruns the selected domain of representation Choose the item Adjust under the Special menu The curve is not defined this means that the equation returns an undefined number for ex dividing by zero log x for x lt 0 To check this choose the item Value under the menu Options and check a few points You should see that the result is undefined Contacting amp Updating Don t hesitate to give us part of your opinions comments or questions This is the way we improve our product Bug reports are also welcome Address Arizona Simon amp Jean Bovet Sablons 35 2000 Neuchatel Switzerland Phone 41 32 725 39 04 E mail arizona curvuspro ch Internet The latest version of Curvus Pro is always downloadable at the following address as well as several tutorial files etc http www curvuspro ch
8. menu Selected curves are marked with a V in the Curve menu and are highlighted in the document window To modify the expression of a curve just e Double click on it in the document or e Select Modify under the Curve menu after the curve has been selected To modify the line attributes of a curve color pattern width use the following icons of the Tool toolbar To delete the selected curve s use the Backspace key To duplicated the selected curve s chose Duplicate under the Edit Modifying the Window Range Zooming In or Out To zoom in or out chose the corresponding item under the Options menu or click on the icons and of the Options toolbar Click then on the point of the document window to zoom at You can also use the icons ull situated in the bottom left corner of the document window Expanding a Rectangular Region To expand a rectangular region of the window range chose Expand under the Options menu or click on the icon L of the Options toolbar Drag then the desired box to be expanded in the document window Is not valid for vector fields scalar fields and logic relations Getting Used in a Few Seconds 5 Adjusting the Window Range To adjust the window range means to change its values so that all the curves fit on the visible part of the window To do so chose Adjust under the Options menu or click on the icon 2 of the Options toolbar This option can be quite useful when
9. File under the File menu Data Array To export a data array corresponding to a curve or a graph into a text file chose Export Data Array under the File menu Select the High Precision checkbox to export the values with a maximum number of digits Otherwise the current number format is used The created file can be then opened by any spreadsheet Advanced Features 37 3DMF File If the QuickDraw 3D mode is activated you can export the graph as a 3DMF model by choosing Export 3DMF File under the File menu The created file can be read by SimpleText e g Sound Curvus Pro provides a sound synthesizer that create sound according toa cartesian curve of type y x Chose Synthesize Sound under the Special menu In the dialog appearing you can change the different parameters Click on Play to play the sound or on Save to save it into a file The created sound can be played again just by double clicking on it in the Finder Annexes 38 Annexes The Toolbars Each toolbar can be shown or hidden from the corresponding item of the Window menu Tools Toolbar Equation Editor Mark Sign to mark a point in the document Background attributes or B zier Curve integrated area Line attributes or object frame Annexes Options Toolbar 2D 39 Zoom In Expand Document Region Adjust Evaluation Find Intersection Snapshot Options Toolbar 3D Zoom Out Move Axis
10. c options animations QuickTime movies QuickDraw 3D provide a powerful way of visualizing the evolution of these objects as function of some parameters Eventually different exporting options are provided e High quality printing e High definition picture exporting including PostScript format etc Getting Used in a Few Seconds Getting Used in a Few Seconds Creating a New Document After starting the application or after selecting New under the File menu following dialog shows up Cancer Select the desired document type by clicking on the corresponding tab Chose then a layout from the list and confirm the dialog 2D Document Note The meaning of the word curve is here quite general it includes any element of a 2D document described by one or several algebraic relations such as ordinary curves vector fields scalar fields and logic relations Creating a New Curve Select New under the Curve menu In the appearing dialog chose the desired curve type and confirm the dialog Note The different coordinate types are described in the chapter Coordinates on page 10 Enter then the expression s describing the new curve in the dialog that shows up and confirm Getting Used in a Few Seconds 4 Manipulating Curves To select a curve just e Click on the curve in the document window with the Shift key down for a multiple selection or e Select the corresponding item under the Curve
11. cted objects Object Move Forward a Mowe to Front i Move Backward Kove to Back t Chose the items Arrow and Specifications to modify the appearance of the selected object Use the items Attach to Real Field and Detach From Real Field to attach or detach the selected objects to or from the real field See the illustration under to understand the difference between an object attached and detached from the real field Specifications Attach To Real Field Detach From Real Field Group Ungroup Laying Out a Document 23 After Expand This object is attached to the real field It stays centered on the point 0 0 75 pee This object is detached from the real field It remains in the lower right corner of the document Difference between an object attached and detached from the real field Use the items Group and Ungroup to group and ungroup the selected objects The Special Menu 2D Document In a 2D document you can add horizontal or vertical lines by choosing the corresponding items under the Special menu and by click on the desired point of the document window To modify the position of an existing line either double click on it or just drag it To insert a text zone containing the value of an algebraic expression updated continuously chose Insert Expression Value and click on the desired point of the document window In the dialog that shows up you
12. d lt 1 min a b c lower value max a b c greater value erf erfc Igamma gammaP gammaQ beta Rounding int floor greatest integer number lower or equal ceil smallest integer number greater or trunc closest integer number starting from zero round rounded round x int x 0 5 Note Various syntaxes are recognized for some functions abs x x sqrt x sqr x Vx fact x x Entering an Expression 16 Note Function name are not case sensitive sin and SIN are two names of the same function Entering an Expression 17 Syntax The syntax of Curvus Pro is the traditional algebraic syntax Moreover Curvus Pro also manipulates complex numbers w x iy or w x jy The part or the result to be considered real part imaginary part modulus or argument can be selected in the pop up menus of the curve definition dialogs see chapter Graph Definition Dialogs on page 43 Simplifications The multiplication sign may be omitted if implicit Examples 2x 2 x mr n r a x a x a x x 2 X 1 x 2 x 1 If a function has only one parameter and if this one does not contain any operator except exponent the brackets can be omitted Examples sinx sin x log2x log 2x cosx cos x tan2x tan 2x Inabsx In abs x In x but sqrt2 x sqrt 2 x Ina b In a b sin x sin y sinx siny sinxsiny sin xsiny If a function is taken to a power following syntax may be used f x
13. desired place to insert the drawn objects into the document For vector and scalar fields the right part of the dialog is a little bit different Only available for cartesian curves in a 2D document Numerical Evaluations 28 Display mrema M Fat TE 2 Vector Pimia Ficech Lites Pree i Click on the Plot a Field Line button to insert a field line starting from the calculated point You can also acheive this by click on the document window with the Option key down Root Finding To find roots extrema or inflexion points of a curve chose Root Find under the Options menu or click on the icon of the Options toolbar Following dialog appears of vasa Piets Tanoent Er yei i i se P har eS ya 3Pe l ya A Te eT Ato ty ai Ooi 4 ye beets iari et yg E Select in the pop up menus of the upper part the kind of find to perform enter a starting value and click on Find Finding Intersection between Curves Chose Intersection Find under the Options menu or click on the icon of the Options toolbar The procedure to follow is the same as the one described above Integrating To calculate the area of the surface under a curve the length of an arc the volume of lateral area of a rotation surface chose Integration under the Options menu or click on the icon of the Options toolbar Following dialog appears This option is only available for cartesian curves of type
14. e corresponding item under the Graph menu The selected graph is marked with a V in the menu Modify ae A Appearance G M To change the expression of the selected graph Remove chose Modify Fex fay v To modify the appearance of the selected graph Vv zer2sin3phi chose Appearance To delete the selected graph chose Remove Modifying the Graph Range Zooming To perform a zoom e Chose Zoom In or Zoom Out under the Options menu e Click on the icons and of the Options toolbar e Click on the icons ula in the bottom left corner of the document window Adjusting the Graph Range To adjust the range to the plotted graphs chose Adjust from the Options menu or click on the icon of the Options toolbar Shifting the Horizontal Range To shift the horizontal range i e along x and y axis chose Horizontal Offset under the Options menu on click on the icon of the Options toolbar Click then on the graph and drag the horizontal range while keeping the mouse button down Getting Used in a Few Seconds 8 Modifying the Values of the Graph Range To modify the minimal and maximal values of the graph range chose Graph under the Format menu or double click on the graph in the document window Click on the Domain tab You can then change the different values and scales linear or logarithmic You can also enter directly these values in the Window Range toolbar chose corresponding item under Window me
15. eate a QuickTime movie from the recorded path Dynamic Options 34 Deactivating Navigation Mode Just click on the close box of the Navigation toolbar or chose Close under the File menu The recorded path is kept in memory and can saved with the document file Note The navigation mode is automatically deactivated when another document is brought to the front Advanced Features 35 Advanced Features Numerical Data In a 2D document you can add a curve defined by a set of numerical data corresponding to an array of values For that chose Numerical Data under the Special menu In the dialog appearing you can enter a set of points or import it from any text file Slide Show A slide show can be performed from a folder containing several Curvus Pro files They are displayed in turn during a certain amount of time Many slide show options are available Creating a Slide Show Create a folder in the Finder in which are placed or saved a set of 2D 3D documents or QuickTime movies Slide Show Options You may define some options for each of these files To change them open the file and chose Slide Show Options under the File menu Important Do not forget to save the document after modifying any slide show options 2D Document The slide show options for a 2D document can be modified in the following dialog Tialay Deration Default 2 i seconds vary Farameber Dede D You can specif
16. ences Dialog This dialog provides numerous options and is therefore quite large Nevertheless the meaning of most options is quite obvious Only the few that may be confusing are described below The generic preferences are used when no document is available or if the current document type is not concerned like the 3BD Document tab for a 2D document See the first line under the tabs Note Click on the Default button to restore the default values for the displayed options 2D Document Click on the 2D Document tab to obtain following dialog Pregeresres OF Lnteted 1 Zoom Factor SOs Austriari es Aima Expo riicey EA FR Teck marks ter i a grid ef about H imini Caves em 3 E lich Paced crit When Copyeey Follow Wines o If oat OF ihe CeO AFG Stee on kanti E solncted incis t By aY Oy Cl Asis Jord E Haireress Smooth Curves If this checkbox is selected the curves are smoothed when they are exported not in high definition This will avoid some steps to appear when the picture is exported toward another application Note The printing quality will be the best if using a PostScript printer thanks to the PostScript data included to the exported picture Follow Mouse When the item Stop on Roots of y is checked and while moving the mouse cursor over a root of a curve in the document window the exact value of the root will be calculated A tick sound is then heard Analogously when y is chec
17. f x Example sin x sinx sin x but sinx sin x Expression Defined by Parts To define a expression by parts Curvus Pro offers various syntaxes condition1 expression condition2 expression2 etc or expression1 conditioni expression2 condition2 etc or expression condition other expression You just have to separate condition and expression or by Return Entering an Expression 18 See following examples sinx X lt 0 x x 1 x gt 0 or X lt 0 sinx x gt 0 x x 1 or X lt 0 sinx X x 1 or SINX X lt 0 x x 1 x lt 1 x or x xe 1 1 The conditions are Boolean conditions The Boolean operators are mentioned in the chapter Boolean Operators on page 11 Examples Ix lt 1 SINX lt COSX X gt 0 amp x lt 2 equiv to x 0 2 or x20 x lt 2 x a b O 0 2 equiv to x e a b A x 0 2 Curvus Pro also treats conditional expressions The syntax is the following IF condition THEN expression1 ELSE expression2 or IF conditioni THEN expression1 ELSEIF condition2 THEN expression2 ELSEIF condition n 1 THEN expression n 1 ELSE expression n Example IF x lt 0 THEN 0 ELSE exp 1 x Using Parameters and Custom Definitions Variable Parameters The user can define variable parameters in Curvus Pro which are helpful to draw a function for different values of this parame
18. h with the mouse by dragging it in the document window For vector or scalar fields this part is a little bit different b Display GA Pes itima Vector GA Riata Piot a Feed Link meea i Click on the Plot a Field Line button to insert a field line starting from the calculated point Dynamic Options 31 Dynamic Options Animations Varying a Parameter Once your 2D document contains one or more families of curves see chapter on page you may display only the curves corresponding to one value of a parameter To do so chose Vary a Parameter under the Curve menu Following dialog appears Vary a Farameber Select the parameter to vary and click on the icon Lr to start the animation or move the cursor to change the value of the parameter Creating a QuickTime Movie In order to visualize the evolution of some curves or graphs as function of certain parameters or when changing the window range you can create QuickTime movies with Curvus Pro Chose Animation under the Special in a 2D document or Options in a 3D document menu or click on the icon or in a 3D document Following dialog appears ar a a J te show iar of inne w packed Opthos Frequency fires Hare equals D as n a a T aneat In the top left part select the parameter s which value is to be varied For that chose the desired parameter or custom constant in the pop up menu check
19. ick on the General tab to display the follow dialog e A thanrerd halogs EA Colors in kienu Cures Conclusion 50 Conclusion Troubleshooting Bad Quality Printing When export an image toward another application and print it the curves show staircase defects This is a problem that occurs in the majority of applications This happens because before being printed all images are converted to 72 dpi leading to a loss of quality for high definition images Look at the examples shown at right these are enlargements e The first curve was printed directly in Curvus Pro without any defects e The middle one has been exported to another application The very annoying staircases show up in the printout even if the curve has been exported with high definition e The last one has been exported with the ae smoothing option and the printed in another application Even so very slight d imperfections persist because the printout is done with 72 dpi and not in 360 dpi like in Curvus Pro the quality is much better than before To solve this problem be sure that the option Smooth Curves in the Preferences dialog see page 47 is activated normally on then copy the image normally without keeping the Option key down You will get the best quality in your printouts Printed in Curvus Pro Exported with the smoothing option Important Avoid reducing the size of an image containing curves after
20. ing an Expression Operators 11 Scalar Operators Here is a list of the recognized scalar operators listed by priority Priority 1 A Priority 2 ns Priority 3 Note power multiplication division modulo addition subtraction Curvus Pro provides a particularly simple syntax to define a function by parts See chapter Expression Defined by Parts on page 17 Boolean Operators Here is a list of the recognized boolean operators listed by priority Priority 1 gt lt 2 gt gt lt lt lt lt gt l NOT NON A AND ET amp v OR OU XOR EOU e IN DANS Priority 2 Priority 3 Priority 4 Interval Operators strictly greater strictly lower greater or equal lower or equal equal different logical no logical and logical or exclusive or inside interval The recognized interval operators are the following Priority 1 Priority 2 n Priority 3 U NOT NON complement intersection union difference A interval is defined with square brackets Examples a b Ja b a b closed interval open interval semi closed interval Entering an Expression 12 Special Operators Sum X Somme or Sum The syntax is the following max Y fvar var min or without equation editor var min max f var Example 10 YE 1 2 10 n 1 Product
21. ked Curvus Pro will round the haircross position to the next extremum of the curve Annexes 48 Haircross Check this option to activate the haircross appearing when moving the mouse cursor over a curve in the document window 3D Document Click on the 3D Document tab to obtain following dialog T h Danem Ti E erre it ral 1 ees esate rer Ii per Preferences Of Unii 2 Zoom anr Adjuv Margin Halow Faces Thickness When Exporting kf Include Posticript Adjuri lick marks far a grid af about oz om 2 O Absolutes Rotations cance C Automatic Proportions Check this option if you want the proportion ratios to be adjusted automatically according to the types of the graphs contained in the document see chapter View on page 25 For example a graph containing one cartesian curve z x y will have following ratios xandy 100 maximum width Z proportionnal Absolute Rotations When this option is checked rotations are done about the absolute axes instead of the graph axes Root Find Click on the Root Find tab to modify the parameters concerning the find of root or intersection only for 2D document Numbers Click on the Numbers to change de default number format Equation Editor Click on the Equation Editor tab to modify the parameters for the equation editor You can enable or disable it select the default font and the key combination for line feeds Annexes General 49 Cl
22. lighting in QuickDraw 3D mode Numerical Evaluations 27 Numerical Evaluations In Curvus Pro you can easily perform numerical evaluations such e Calculating the coordinates of a point from a graph its various derivatives tangents and osculating circles etc e Finding roots extrema and inflexion points e Finding the intersection of two cartesian curves e Integrating the area under a curve the volume of a graph etc Note It is possible to enter a expression in any field where a real number is required e g exp2 27 14 V5 2 etc 2D Document Evaluating Values at a Point To evaluate different values at a point of a curve select Evaluation under the Options menu or click on the of the Options toolbar Following dialog appears Smet ita 4 j Display e aS 7 Beana ee Blangent B yal DA Follow eee A tulaten rrie Cente 1 Se ei 2 Packt i Select in the top left pop up menu the desired curve Fill in the field s under it and click on Calculate It is also possible to move the mouse cursor over the document window if the option Follow Mouse is checked The values of the several coordinates derivates etc appears in the middle box In the right hand part of the dialog which can be masked or shown by clicking on the little green triangle it is possible to select the elements to be shown in the document window Click on the Plot button or click in the document window at the
23. nge the document margins chose Document under the Format menu and click on the Margins tab Laying Out a Document 25 Modifying the Graph Appearance in a 3D Document To modify the graph appearance chose Graph under the Format menu or double click on the graph in the document window Following dialog appears Graph Format O lagartas Scar Younis ye O Departihmic Scale k O lepar hmir Scale Graph Range Click on the Domain tab to change the graph range as well as the different scales View Click on the View tab to change e The angle of view by entering the rotation values e The perspective kind e The three axis ratios Proportional means a width of the graph proportional to its range as the percentage indicates a fixed width Bounds Click on the Bounds tab to select e The different colors of the graph faces x 0 y 0 and z 0 planes e The background color e The three legends Axis Click on the Axis to modify e The three axis e The tick marks and numbering Laying Out a Document 26 e The main and second grids Note The tick mark spacing value is automatically set according the value entered in the Preferences dialog see page 46 Number and Font The two tabs Numbers and Font reveals options to change the format and appearance of the numbering See page 24 for the details about these options Light Click on the Light tab to modify the light source or the
24. nu to show it Miscellaneous Items of the Options Menu To center the axis origin chose Center Axis To make the basis orthonormal chose the corresponding item Numerical Calculation To evaluate different values at a point a of graph coordinates derivatives etc or to integrate a function chose Evaluation or Integration under the Options menu or click on the icon of the Options toolbar For further information see chapter Numerical Evaluations on page 27 QuickDraw 3D When running Curvus Pro on a Power Macintosh with the QuickDraw 3D extension enabled the QuickDraw 3D mode can be activated by chosing the corresponding item under the QuickDraw menu This mode allows the user to manipulate the graph in real time or to navigate around it by moving the viewpoint For further information see chapter Navigation on page 32 Entering an Expression 9 Entering an Expression The Equation Editor Curvus Pro provides a user friendly equation editor ideal for entering and manipulating mathematical expressions Its ease of use allows any user to rapidly get advantage of it The Equation Editor Toolbar To insert special characters or different symbols use the toolbar appearing at the top of the screen just under the menu bar Click on a button to insert the corresponding character or symbol Click on an arrow to pull down a pop up menu proposing other symbols The most recently used symbol is sho
25. oselect a definition click on it Comment To select several definitions keep the Shift key down or Command key for discontinuous selection Save the definitions Click on this icon to save the definitions in a file Load the definitions To modify a definition double click on it Click on this icon to load To copy into the clipboard the expression definitions from a file of the selected definitions choose Copy in the Edit Menu Increment bar Move a definition Click on on arrow to move a definition in the list in order to Select a definition typically a constant have it sorted in the desired and click on this bar to increment or order decrement its value Note See chapter Custom Definitions on page 19 for further information about using such custom definitions Window Range Toolbar Enter the boundary values of the window range Click on the triangle to restore a recently used range Annexes 42 Calculator Toolbar Insert a pre defined function or Mode constant Choose the mode degrees radians grades for the Confirm Computes the expression and shows the result trigonometric Equivalent to typing calculations Return or Mode degrees C145 2 1 618 Expression Click on the zone to select the text type in the expression you want to evaluate and confirm To use the previous result click once more on the right of the sign and type in the new expre
26. rs expression Examples area radius n radius mean p q p q 2 Note Custom definitions may have references to others Plug Ins Curvus Pro let you create your own plug ins A plug in is a compiled routine which defines a function that cannot be expressed algebraically Example The plug in Mandelbrot returns the maximum iteration number so that z lt 2 the iteration being defined by Z 2 G c atib z 0 See the file How to create your own plug ins for further information on this subject Entering an Expression 20 Note Plug ins easily spotted by their icon igh must be placed in the folder Plug Ins located next to the application Laying Out a Document 21 Laying Out a Document Document Format To change format and dimensions of your document chose Document under the Format menu Following dialog appears a hw fice Margins E Pindiy SU Dassic oO L 0 amy ine Layouts A document layout contains several parameters of the different formats of axis and grid for 2D documents or of the graph for 3D documents Many layouts are available when creating a new document see chapter Creating a New Document on page 3 To create a new layout from the current document click on New You can apply the selected layout to the current document by clicking on Apply To change an existing layout according to the current document click on Modify Eventually you can rename or erase a layout
27. s C Mw CF Coshees Delinttiees Near Double click on a element of the list to modify it See chapter Using Parameters and Custom Definitions on page 18 for further information Annexes 45 Miscellaneous Options The bottom part of the dialog provides miscellaneous options specific to each kind of graph Following options are the most frequent Activate this option to better Mode for plot the curve at its extremities trigono metric calculations T tiei aE Minda Fii a Best Extremities omple Read Fari O Ms ni Alasisy during Enei Partof the result to be displayed when using complex numbers Specific Options Cartesian Curves y x Vertical Asymptote If this option is activated the curve is plot with a continuous line Otherwise vertical segments corresponding to vertical asymptotes or jumps are not drawn See the example below rit M aia te without item unchecked with item checked Example of plot with and without the option Vertical Asymptotes Annexes 46 Differential Equations Initial Conditions That are the boundary conditions for the initial time usually zero Diferenteal Equateces k ija Conditeans bt LPredeied Perwiien El i a a Note For a system of differential equations of order two or greater you can use the variables x and y or x et y as well as v V x 24 y the velocity Annexes 47 The Prefer
28. some curve exceeds the current window range and is so not or only partially visible Other Items of the Options Menu To shift the window range select Shift and drag the axis origin To center the axis origin to the middle of the window chose Center Axis To make the basis orthonormal i e same scale for both axis chose Make Basis Orthonormal To modify the tick mark spacing and so the range of the corresponding axis chose Extend Tick Marks click on a tick mark and drag it Modifying Window Range Values Chose Document under the Format menu and click on the Domain tab You can then enter the different minimal and maximal values for the window range select the scales linear or logarithmic as well as the reference frame 1 unit 1 cm on one axis e g Select the Orthonormal Basis checkbox to keep the document basis orthonormal when modifying the window range You can also double click on one axis and select the Range tab to modify the values of the corresponding axis range You can also enter directly the different values in the Window Range toolbar chose the corresponding item under the Window menu if hidden Copying into the Clipboard To copy the whole picture of the document window into the Clipboard chose Copy from the Edit menu no curve must be selected To copy only a part of the window chose Snapshot under the Special menu or click on the icon le of the Options toolbar Drag then the desired box in the doc
29. ssion Note All calculations are performed with complex numbers either i or j is recognized for the imaginary part Annexes 43 Graph Definition Dialogs Most curve and graph definition dialogs have a similar common structure which main features are described below For some special graphs the dialog presents some specific options that are described starting from page 45 When a new curve is created or when an existing curve is modified a dialog similar to the one below appears f fret b Predefiend herein mepe oe k Peedethic rs ACuatoin bans hte eae Note Click on the green triangles to expand the dialog and show further options Entering an Expression In the upper part of the dialog enter the expression s describing the graph For some graphs you may also enter the range of the parameter s Note You may enter an expression in any fields that requires a numerical value For example you can define a polar curve for between 0 and 27 You may even make reference to a parameter in any of these fields Annexes 44 Click on this triangle to restore a recently used axnressian rtp ie hta Pree fae Perci ier Farameter Click on this triangle to insert a predefined constant of function Parameters and Custom Definitions In the middle part you can create modify or delete a parameter or a custom definition 7 Parameters Custom Medinet ores Parameter
30. stants true vrai Boolean yes result of 0 0 false faux Boolean no result of 0 1 nil nan nota number result of In 0 1 0 etc Recognized Variables The following variables are recognized by Curvus Pro X Y Z r 0 0 coordinates see chapter on page 10 t u parameters for parametric curves v speed r for differential equations Note Constant and variables names are case sensitive r and R are names of two different constants Recognized Functions The following functions are available in Curvus Pro Trigonometric sin cos tan cot ctn sec CSC asin arcsin acos arccos atan arctan acot arccot actn arcctn deg value in degrees of the angle as parameter rad value in radians grad value in grades hypot hypot a b saqrt a b Entering an Expression 15 atan2 atan2 y x returns the arctangent of y x in 2 2 according to the signs of x and y Hyperbolic sinh cosh tanh coth asinh arcsinh argsinh acosh arccosh argcosh atanh arctanh argtanh acoth arccoth argcoth Bessel I n x J n x K n x Y n x Complex Numbers cis cisB cosB isinB re real part im imaginary part arg argument in 7 7 Miscellaneous abs exp fgamma fpart fractional part int ipart integer part inv inverse log basis 10 In natural sq square nsqrt n x n th root of x sqrt sqr Square root sgn sign fact factorial rnd random number O lt rn
31. ter For example you can plot the function x kx 1 for k 1 1 2 1 4 et 2 In order to define them there are two possibilities e Use the Parameters toolbar see chapter Parameters Toolbar on page 40 e In adialog of curve definition select Parameters after having clicked on the first green triangle click on the New button see chapter Graph Definition Dialogs on page 43 Entering an Expression 19 Once the parameter is defined you can create any graph with an expression containing the name of it Example e Define variable n which vary between 1 and 3 in 5 steps of 0 5 e Create then a curve of equation sin nx e Five curves are plotted for the values 1 1 5 2 2 5 and 3 of n Note An expression cannot make reference to more than one parameter at a time Custom Definitions In Curvus Pro you can define your own constants or functions You have two possibilities for creating your own definitions e Use the Custom Definition toolbar see chapter Custom Definitions Toolbar on page 41 e In adialog of curve definition select Custom Definitions after having clicked on the first green triangle click on the New button see chapter Graph Definition Dialogs on page 43 You only need to enter the expression of the constant or function in one of the following forms Constant name expression Examples diameter 2 4 perimeter x diameter phi V5 1 2 Function nam paramete
32. the box next to it and enter the initial and final values Select the checkbox Dynamic Options 32 Show if you want the parameter value to be shown at the bottom of the animation In the other fields you may enter different animation parameters frame number rate size compression etc In the bottom part of the dialog you can chose the initial and final window range as well as the view angles in a 3D document Confirm the dialog to create the QuickTime movie You may also create QuickTime movie from a navigation path See chapter Navigation on page 32 for further information Using QuickDraw 3D QuickDraw 3D is a system extension providing three dimensional object visualizing and handling Curvus Pro take advantage of it for real time 3D graph manipulation Note This extension is not available for 680x0 based machines To activate this display mode of course only available for 3D documents select Activate QuickDraw 3D under the QuickDraw menu This mode is used exclusively for the graph display so that no vector drawing isn t shown nor can be used Important Be sure that this extension is located in the Extensions folder of your System Folder Navigation Once QuickDraw 3D is activated chose Navigation under the QuickDraw menu to swap to the navigation mode Follow toolbar appears then click on the green triangle to show or mask the right part Hel eta A boo i nd A
33. ument window To copy a picture containing only some curves select the curves to be included in the picture and chose Copy under the Edit menu see chapter The Preferences Dialog on page 47 Note To copy a high definition picture keep the Shift key down while choosing Copy under the Edit Click and grow box while keeping the mouse button Getting Used in a Few Seconds Numerical Calculation To calculate the coordinates of a point from a curve shown in the window just move the mouse cursor over it a haircross appears which coordinates are shown at the bottom of the document window Sans tere I To evaluate different values at a point of a curve coordinates derivatives tangents etc to find a root or an intersection of cartesian curves or to calculate an integral select the items Evaluation Root Find Intersection Find or Integration under the Options menu You can also click on the icons oT H E lt or SI of the Options toolbar For further information see chapter Numerical Evaluations on page 27 Getting Used in a Few Seconds 7 3D Document Creating a New Graph Select New under the Graph menu In the dialog showing up select the type of graph you want to create and confirm Enter then the expression s describing the new graph in the next dialog appearing Manipulating Graphs The default selection is the most recently created graph To select some other graph select th
34. wn in the corresponding button Note See chapter The Preferences Dialog on page 48 to activate or deactivate the equation editor and to select the default police Moving the Cursor Use the arrow keys to move the cursor in a field as well as reaching numerator or denominator of a fraction limits of a sum 20 a product or integral X n 5 To expand the selection keep the Shift key down To Ik move to the next term keep the Option key down kel Use the Tab key with Shift key down to move to the next previous field of a fraction sum integral etc n 0 Brackets roots and fractions To put some text in brackets or into the numerator of a fraction make it the selection and type resp 1 z 1 8 Entering an Expression 10 To remove a symbol such as a root a fraction a sum etc place the cursor at the left of the main field denominator for fraction and type Backspace WH E Drag amp Drop If the Drag amp Drop extension is available on your system you can easily move the selection or copy it when Option key is down by dragging it with the mouse cursor Coordinates Coordinates in the Plane Curvus Pro uses two kind of planar coordinates y P Polar Cartesian Coordinates in the Space Curvus Pro uses four kind of spatial coordinates Z lt N X Polar Cartesian r x QO Cylindrical N ora Spherical Enter
35. y a custom display duration in the top left part Click on the Image button to add a title picture which is shown under the document during the slide show Select the Vary Parameter checkbox to let the value of the selected parameter being varied during the slide show see chapter Navigation on page 32 Advanced Features 36 3D Document The slide show options for a 3D document can be modified in the following dialog aplay Deraijon Er Defautt a SE Domes e Dis taynando orm E Fesjiipa Beginning De To Ceme ox The options of the upper part are the same as the ones for a 2D document On the other hand you may chose between three display modes e Normal the file is simply opened e Transition during the slide show a transition is performed between to selected viewpoints These can be defined in the lower part of the dialog Click on Define to change a position according to the current point of view e Navigation the recorded navigation path see chapter Navigation on page 32 will be followed during the slide show Starting the Slide Show To start the slide show chose Launch Slide Show under the File menu In the dialog appearing then chose the folder containing the files to be shown and confirm it During the slide show you can skip to the next document with the arrow keys Exporting PICT File To export the picture of the current document into a PICT file chose Export PICT
36. y x Numerical Evaluations 29 yesin Method of ft ate usec Select in the upper part the graph the kind of integration to perform and the method to be used Fill in the two integration limits click on the triangles for infinite limits and click on Calculate You can calculate the area between two curves by selecting the checkbox Difference Select the checkbox Draw Area if you want the calculated area to be drawn in the document This latter can be removed by chosing Remove Integrated Area under the Special menu the curve must be selected Numerical Evaluations 30 3D Document Evaluating Values at a Point To evaluate different values at a point of a graph chose Evaluation under the Options menu or click on the icon of the Options toolbar Following dialog apears spay GA Peach ite A Tangent in x Tangent in y Piinia Ela E etic Li A Sormale i ao 4 Gradient Ovecteurs Select in the top left pop up menu the desired graph surface field or curve Fill in the following fields and click on Calculate The calculated values coordinates derivatives etc are shown in the middle box Important The following options are only available when the QuickDraw 3D mode is activated In the right hand part of the dialog which can be masked or shown by clicking on the little green triangle it is possible to select the elements to be shown in the document window You can also spin the grap
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