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Gaigen User Manual Version 0.99

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1. e3ga amp scp const e3ga ta const e3ga amp b 3 4 4 The Inner Products The are currently four inner products available in Gaigen two of which have no operator symbol The number of inner products arises from the lack of con sensus in the geometric algebra research community of which inner product should be preferred The definitions for the inner products are 3 5 INVERSION 27 left contraction e3ga amp e3ga operator lt lt const e3ga amp a const e3ga amp e3ga operator lt lt const e3ga amp a e3ga amp lcont const e3ga amp a const e3ga amp b right contraction e3ga amp e3ga operator gt gt const e3ga amp a const e3ga amp e3ga operator gt gt const e3ga amp a e3ga amp rcont const e3ga amp a const e3ga amp b Hestenes inner product e3ga amp hip const e3ga amp a const e3ga b Modified Hestenes inner product e3ga amp mhip const e3ga amp a const e3ga b 3 5 Inversion If three inversion functions are available versor lounesto and general inverse then which one does Gaigen pick when you write one of the following equiva lent line of code a b a a b c a b c inverse The answer is that Gaigen will prefer the versor inverse over the lounesto inverse and the lounesto inverse over the general inverse So if you have included all three inversion functions in your algebra Gaigen will automati cally use the fastest versor inverse If the vers
2. A e2 A e3 The GRADE1 macro tells the constructor that the coordinates specified are for grade 1 of the multivector The GRADE3 macro tells the constructor that the coordinate specified is for grade 3 of the multivector Note that the order of coordinates matters here It depends on the order you specified in the order tab in the user interface when you created the algebra For instance if you wanted to create a multivector with the value of le A e2 2e1 A ez 3e2 A ez you would have to use e3ga c GRADE2 3 0 2 0 1 0 because the e3ga algebra stores its grade 2 coordinates in the following the order and orientation ez e3 ez e1 e e2 You can inspect and change this order and orientation by loading the specification file e3ga gas for the algebra into the Gaigen UI click on load algebra and clicking on the order and g 2 tabs Of course you can also construct such an object explicitly from basis vector as follows a e3ga c 1 0 e3ga e1 e3ga e2 2 0 e3ga el e3ga e3 3 0 e3ga e2 e3ga e3 This removes all doubt but is less efficient In this code snippet we have used some features we have not discussed yet such as the and operators and the use of e3ga el to denote the e basis vector We will treat them later Now suppose you want explicitly set the coordinates of multiple grade parts of a new multivector variable You can do that like this float coordinates 4 1 0 2 0 3 0 4 0 e3ga d
3. e g e that de scribes how to format floating point coordinates The second function takes as arguments to characters e g and that will be used as start and end of every string Two functions and one operator provide direct read only access to the coordinates of a multivector scalar coordinates and The scalar function returns the scalar coordinate of a multivector and is used like this e3ga a 1 0 float sc a scalar The coordinates function and the operator can retrieve a pointer to the coor dinates of any grade part of a multivector You specify which grade part you ll get the the coordinates using the integer argument This argument can be any of the GRADEO GRADEN macros but you can not combine then to get the coordinates of multiply grades in one call This example demonstratres the use of the coordinates function and the operator e3ga a GRADE1 1 0 2 0 3 0 get a pointer to the grade 1 coordinates of a float glc a coordinates GRADE1 get the scalar coordnate of a float sc a coordinates GRADEO 0 get each of the grade 2 coordinates of a float g2c a coordinates GRADE2 E E f loa g2c el e2 g2c 2 g2c 0 g2c_e3_e1 g2c 1 loat g2c_e2_e3 loa 3 13 COORDINATE OUTPUT AND ACCESS 35 In the last four lines of the example above you can how the each of the three coordinates of the grade 2 part of a is retrieved This piece o
4. 0 or a versor versor 1 The function returns the number of factors and stores them in f int e3ga factor e3ga f int versor 0 const Factor a versor The function returns the number of factors and stores them in f int e3ga factorVersor e3ga f const 3 11 Exponentiation The function exp which computes the exponentiation of a multivector is present in the algebra whenver the geometric product is included The function com putes the taylor series expansion n exp A Y E 3 1 1 0 n is specified by the optional integer argument which defaults to 9 This is identical to the CLU exp function The definition of exp is e3ga amp e3ga exp int order 9 const 3 12 Outermorphism If you have a function f for which the following is true for any pair of input blades a and b f a b fla A f b 3 2 then it is an outermorphism Examples of outermorphisms are rotations An other example are all operations that you traditionally can model using a 4 x 4 when you use homogeneous coordinates translation rotation scaling skew ing and so on If you have to apply such a function to multivectors many times you might consider constructing an outermorphism for it It can be ap plied to blades of any grade Once initialized the outermorphism operator can probably compute the result faster and will assure that the result is the same grade as the input The outermorphism operator is
5. 0f The distribution of the random value is linear from the range 1 0 1 0 The range is controlled by the second argument To set a multivector variable to a higher grade value the following would be used b randomBlade GRADE2 5 0f c randomVersor GRADE3 0 5f This sets b to a grade 2 blade with a random value and c to a versor with the highest non empty grade part 3 The blades and versors are created by gener ating the required number of random vectors within the range specified by the second argument and respectively wedging and multiplying these together Higher grade random blades and versors are thus not taken from a linear dis tribution The definitions of all of these assignment functions are set to 0 void e3ga null set single grade with N coordinates int e3ga set int grade float cl float cN set multiple grades void e3ga set int gradeUsage const float coordinates set single grade with N seperate coordinates int e3ga setScalar float cl int e3ga setVector float cl float cN1 int e3ga set2Vector float cl float cN2 int e3ga setNVector float c1 set single grade with an array of coordinates 3 3 OPERATORS 25 int e3ga setScalar const float coordinates 1 int e3ga setVector const float coordinates N1 2 int e3ga set2Vector const float coordinates N2 int e3ga setNVector const float coordinates 1 copy a
6. GRADEO GRADE2 coordinates This creates a multivector variable d with the value 1 0 2e2 A e3 3e3 Ae de eg The macros GRADEO and GRADE2 can be added together using the standard binary or operator telling the constructor that you will supply coordinates for the grade 0 and grade 2 parts The coordinates have to be sup plied in an array because supplying a separate constructor for almost every grade combination would grow out of hand If the supplied array is too short to contain all coordinates behaviour of the constructor will be unpredictable There is one more constructor the copy constructor is used to create a new multivector variable with the same value as another variable The following code creates a variable e with the same value as d e3ga e d 3 2 ASSIGNMENT 23 The definitions of all of these constuctors are null constructor e3ga e3ga copy constructor e3ga e3ga e3ga amp a set to scalar value e3ga e3ga float scalar construct amp set single grade with N coordinates e3ga e3ga int gradeUsage float cl float cN construct amp set multiple grades e3ga e3ga int gradeUsage const float coordinates 3 2 Assignment To set an existing multivector variable to the value 0 the function null can be used a null To set the coordinates of an existing multivector variable explicitly you can use one of the set functions They are available in two
7. INTERFACE a b which is equal to a a b which in turn is equal to a copy add a b 3 4 The basic products We use the name basic products for all products which are included using the products tab of gaigenui Thus this definition excludes the delta product the meet and join multiplying by the inverse and other derived products 3 4 1 Geometric Product The function gp computes the geometric product of two multivector variables The symbol is used as the operator for the geometric product The definitions of the geometric product functions are e3ga amp e3ga operator const e3ga ta const e3ga amp e3ga operator const e3ga ta e3ga amp gp const e3ga amp a const e3ga amp b 3 4 2 Outer Product The function op computes the outer product of two multivector variables The symbol is used as the operator for the outer product The definitions of the outer product functions are e3ga amp e3ga operator const e3ga amp a const e3ga amp e3ga operator const e3ga ta e3ga amp op const e3ga amp a const e3ga b 3 4 3 The Scalar Product The function scp computes the scalar product of two multivector variables The scalar product returns the grade 0 part of the geometric product The symbol is used as the operator for the scalar product The definitions of the scalar product functions are e3ga amp e3ga operator const e3ga ta const e3ga amp e3ga operator const e3ga ta
8. When you push the button you are prompted to select a gap Geometric Algebra Profile file These files are writ ten by the saveProfile function see section 3 16 Gaigen will then automati cally add optimizations for all product multivector combinations that are used more than 2 0 of the time You can use the optimize threshold in usage per centage slider to change this value of 2 0 Le 0 0 would add optimizations for every product multivector combination that is used in your application If you tick the inline products checkbox all generated products will be prefixed with an inline statement This might improve performance a few percent You can use the Dispatch method radio buttons to select what dispatch method to use As shown in 1 the ifelse method is usually fastest followed by switch 2 4 Order Using the order tab figure 2 2b you can modify the order in which the coor dinates referring to basis blades are stored and you can change the orientation of the basis blades This part of the user interface is a bit primitive though functional You are concerned mostly with coordinates when you enter them into a multivector object e g by using the set function when you retrieve them from a multivector object e g by using the coordinates function or when you in spect them e g by using the print function 12 CHAPTER 2 THE USER INTERFACE General signature products order funcbons storage
9. algebras with the same name but with slightly different properties You would store the algebras in seperate directories e g e3ga app1 and e3ga app2 and compile and link each application with its own algebra which was tailored to its own specific needs The checkboxes at the lower part of the general tab control what kind of optimized code is generated to implement the computation of the products Only C C2 and LAPack are available currently Moreover LAPack is more like 18 CHAPTER 2 THE USER INTERFACE a test implementation to compare the performance of the C implementation to a brute force LAPack approach In the future SSE and 3DNow support may be added The C code generator generates slightly more a few percent efficient code than the C2 code generator However the code generated by the C2 generator might be much smaller especially when you add lots of optimizations in the products tab or when you generate a high dimensional algebra When you have finished an application you may want to try the other generator to see which one gives the best results Chapter 3 Layer 2 high level C interface The most convenient way to use the source code generated by Gaigen is the high level C interface It consists of two classes one for mulivectors one for outermorphism operators which sit on top of the actual code generated by Gaigen The interface is contained in two files gaigenhl h and gaigenhl cpp which you can easily cha
10. below the two sets of checkboxes Then you would check 1 and 3 in the left set the product Rv will have and non empty grade 1 and 3 and you check 0 and 2 in the right set for the inverted rotor Adding this combination will optimize the product Rv R To remove any combination already added you simply click the remove button for that combination 2 3 1 Automatic optimizations using profiles and optimization options Of course once your application gets more complicated you won t be able to tell easily what combinations of products and grade usages you use most of ten That s why Gaigen can include profiling code in your application You can enable profiling by ticking the enable profiling checkbox in the optimizations tab in the products tab This tab is new in Gaigen version 0 95 The profiling code usage will be explained in the next section will count how often you use each combination and on request print or save a list of most used combina tions You can then use this list to manually optimize your algebra but more conveniently you can let Gaigen handle the optimizations for you First of all you can use the remove all optimizations button to remove all product optimizations from the algebra specification This is recommended before adding automatic optimizations to make sure no old optimizations are left behind Then you can use the automatically add optimizations from profile but ton to add optimizations automatically
11. combinations you use and how often you use them With this code included you can run your program and at any time request a dump of that information and use it to enable specific optimizations either by hand see section 2 3 The user interface consists of a number of tabs at the top of the window a number of buttons at the bottom and a large field displaying the contents of the currently active tab in the center Clicking on a tab will raise its contents The contents of each tab allow you to change a specific set of properties of the algebra that will be generated by clicking on the generate button in the lower left of the window The contents of each tab and the properties they control is discussed in the following sections 2 1 General The contents of the general tab shown in figure 2 1a control the dimension the name of the class and source files whether the high level C interface will be included in what directory the source files will be written and the what type of low level computational code will be generated The dimension is set to 0 by default and can be changed by using the drop down box Only dimensions 0 to 8 are allowed because the approach used 8 CHAPTER 2 THE USER INTERFACE fal Geometric Algebra Implement ation Generator version 0 05 general signature products order functons storage generate general signature products order functions storage generate gmenson 3 a enel O rec
12. high level C interface which is what most people will do When you write an C expression such as a b c d temporary variables are used to store the intermediate results i e b c and b c d A method exists to allocate these variables very quickly compared to the default method used by the C compiler The down side of this method is that it allocates the temporary variables from an array with a fixed e g 64 variables size When it comes to the end of the array it cycles back to first entry whose contents will be overwritten with the new intermediate results If your program still had a reference to 2 6 MEMORY 15 2 6 this variable as the intermediate result of a previous computation this refenence will be useles Using the reference as if it still contained the old value will cause your program to malfunction but no crash Check ing the fast temporary variables checkbox will make your algebra about twice as fast but you will have to make sure you don t run out of tem porary variables The main rule of thumb is never to pass references to temporary variables to functions Instead of writing something like e3ga b c d someFunction b c d write something like e3ga a b c d a b c d someFunction a This will make sure you don t pass references to temporary variables to other functions and prevent most problems The only other ways you can get into trouble with fast tempor
13. other functions or products outermorphism Use this function to create an outermorphism opera tor If you have a linear function you can construct an outermorphism operator of it Applying the outermorphism operator to multivector has several advantages efficiency precision floating point noise over sim ply applying the original function to multivector Requires take grade and negate function and the outer product spinor product Including the spinor product function will add special code for constructing outermorphism operators from spinors This func tion will be removed from Gaigen in the future since it was superceded by the outer morphism Requires take grade and negate function and the geometric product and outer product factor blade versor This function factors blades and versors into ar birary vector factors It is used to compute the meet and join Requires norm project versor inverse and take grade function and the outer product geometric product and left contraction meet and join This function computes the meet and join of blades Re quires factor blade versor project and reject These functions project and reject blades onto from blades and versors Requires the versor inverse function left contraction and for projection onto versors the geometric product random blade versor This function generates random blades and ver sors fast temporary variables This function is only of interest if you use the
14. prints out all product multivector combinations to the stan dard output It has an optional argument which defaults to 2 0 which spec ifies up to what usage percentage the product multivector combinations are shown printProfile computes how often each product multivector combina tion is used relatively and if a product multivector combination falls below the usage percentage threshold it is not shown So to print the usage of all product multivector combinations use 3 17 BASIS VECTORS AND INVERSE PSEUDOSCALAR 39 e3ga printProfile 0 0 where the 0 0 tells the function to show all product multivector combinations The default argument e3ga printProfile which is equivalent to e3ga printProfile 2 0 will print the product multivector combinations with a usage above 2 0 saveProfile saves the profile to gap file which can be read back by the Gaigen user interface to automatically add optimizations section 2 3 Use the function as follows e3ga saveProfile profile name gap The definitions of resetProfile printProfile and saveProfile are static int e3ga resetProfile static int e3ga printProfile float threshold 2 0 static int e3ga saveProfile const char filename NULL 3 17 Basis Vectors and Inverse Pseudoscalar The internal class see section 4 contains the multivector variables with the val ues of the basis vectors and inverse pseudoscalar as static members These are also a
15. value to this ben int addName const char name const e3gai amp mv remove a basis element name from this ben int removeName const char name look up a basis element name and its value from this ben int lookupName const char name e3gal amp mv const remove all basis element names int removeAll set the start and end delimiters e g and this can also be done using the int setDelimiters char startDelimiter char endDelimiter set this to defaults no extra names no delimiters int setDefaults 3 15 MISCELLANEOUS FUNCTIONS 37 Usage example e3ga a e3ga_ben eb 1 int nb eb addName I e3ga el e3ga e2 e3ga e3 nb a parseString 1 0 1 amp eb A future version of Gaigen may contain a better multivector parser that uses ANTLR much like the parser that is already present in GAViewer 3 15 Miscellaneous functions 3 15 1 normal The normal function returns a normalized version of the multivector it is ap plied to e g e3ga a b I fies b a normal set b to the normalized version of a The normal function takes and optional integer argument which specifies the norm to be used If the argument is 1 the Euclidean norm is take the sum of the square of all coordinates If the argument is 2 the norm is computed as GAIM_ FLOAT norm a a reverse 3 15 2 dual The dual function returns the dual with respect t
16. 3 are NULL The opt file contains the specifications of the dynamic part of the interface It specifies the name of the algebra with an i appended to it since this is an in ternal part of the algebra implementation the dimension the floating point type and the optimized product functions which should be implemented The opt file format is described in section 6 2 The general product functions which can compute the product of any type of multivector with any other type of multivector always look something like this void e3gai_general_gp const float a const float b float c This function should compute a product of multivector a and b and store it in c The coordinates of the multvectors are given in the format explain above and illustrated in figure 5 1 The optimized product functions which can compute only a product of two specific types of multivectors always look something like this 53 void e3gai_opt_0A gp 05 const float a const float b float c The functions receive the coordinates of their arguments a b and c as arrays of floating point variables The opt2X compiler knows what it should do with each coordinate because of the description of the function in the opt file An opt2X compiler is free to implement all these function anyway it wants The opt2c compiler for instance take the most straightforward approach and almost directly converts the opt file to C The opt2LAPack compiler an experimen
17. Gaigen User Manual Version 0 99 Dani l Fontijne University of Amsterdam November 24 2003 Contents 1 Introduction 5 2 The User Interface 7 2 14 Generalidad A E tea AE a 7 22 Signataire ml daa na ae 9 ZA Products 2 e o do a e a 9 2 3 1 Automatic optimizations using profiles and optimiza Horn options 4 42 a a pie a o a a e 11 DA Ordera oa an A A e er kB 11 2 5 FUNCHONS iae os ta bed ad a ra a 12 226 Memory it da ti at da ty Ae ate e Te 15 2 7 Generates a wo OE A AA A tt 17 3 Layer 2 high level C interface 19 FL Constructora 21 3 2 Assignment 2 ee 23 3 3 Operators io A A ee REE AE A artes 25 3 4 The basic products 2 0 0 00 E eee eee 26 3 4 1 Geometric Product 0 000000 e 26 3 42 Outer Prod cti s 23 scot eS we ee e eae le as 26 3 4 3 The Scalar Product 0 000 0004 26 3 4 4 The Inner Products 0 00000048 26 3 5 TiniVersion s a aak e Ea e ar ae ae I a ees 27 3 6 Addition subtraction negation sosoo 28 3 7 Reverse Clifford Conjugate and Grade Involution 29 3 8 Grade PartSelection aa oe a a E ai eee eee eee 29 3 9 Meetand Joli paani ald ad ly Parte E E BiG gee 30 3 10 Factorization sa o ae eee Le A a a a 30 3 11 Exponentiation 2 nsan ma ap a e eee 31 3 12 Outermorphism sou soa eta o eA Eee ee 31 3 13 Coordinate Output and Access 1 2 2 ee o o 33 3 14 Coordinate string parsing_ o o e 36 3 15 Miscellan
18. SS Chapter 5 Layer 0 low level computational functions Layer 0 is responsible for actually computing the optimized products and computing other basic functions such as addition and reversion It is de fined as an partially static and partially dynamic interface which must be im plemented An opt2X compiler is responsible for generating the layer 0 files although you can of course them by hand if you want The compiler takes an opt file see section 6 2 which contains specifications of the functions that should be implemented and generates a C C or assembly file which adhers to the interface The static part of the interface currently consists of the following functions copy sre to dst length of arrays is length void e3gai_copy float dst const float src int length compute reverse multivector a void e3gai_reverse float all compute clifford conjugate of multivector a void e3gai_cliffordConjugate float al compute the grade involution of multivector a void e3gai_involution float a negate sre to dst length of array is length void e3gai_negate float dest const float src int length compute the norm all coordinates squared of a length of array is length float e3gai_norm_a float a int length add two multivectors with the same grade part usage la lal h length of arrays is length vo
19. The difference is that an array of floating point variables inside the class always uses a certain amount of memory while a pointer can be used to point to any amount of memory So with the array of floating point variables inside the class the e3gai class will still use 8 or 4 floating point vari ables to store a single coordinate e g a scalar while the pointer can point to a much lower amount of memory this is done by the tight and balanded memory allocation algorithms For coordinate access there is no difference between the pointer and the array The coordinates are always stored in compressed form in the c array Even if 8 floating point variable are allocated anyway only 4 of them are used to store for example a rotor The usage integer variable is to store the grade part usage as a bitfield and the memory usage as an integer These two n bit words are stored as illustrated in figure 4 1 If a grade part is in use by the multivector the bit in the grade usage field will be set to 1 otherwise it will be set to 0 The memory usage field stores the number of floating point variables allocated for this mul tivector It is used by the balanced and tight memory allocation algorithms to determine from which heap they allocated the coordinate array The other storage variables of the C class are static which means that only one copy of them is used in your application They include dim the dimension of the algebra nbCoo
20. You can change this format by specifying it as the second argu ment of print e g e3ga a GRADE1 1 0 2 0 3 0 a print a Se This prints a 1 000000e 000 el 2 000000e 000 e2 3 000000e 000 e3 You can also change it using the setFPPrecision function described below If you want to print to something else than the standard output you can use the fprint function to print to any file For total control of where your output goes you can print the coordinates to a string using the string function You can then do whatever you want with that string The definitions of these three functions are 34 CHAPTER 3 LAYER 2 HIGH LEVEL C INTERFACE get a string representation of a multivector const char e3ga string const char prec NULL const print a string representation of a multivector to the standard output void e3ga print const char text NULL const char prec NULL const print a string representation of a multivector to a file void e3ga fprint FILE F const char text NULL const char prec NULL const Starting with Gaigen 0 99 two more function are available that control the printing set the default precision for all future prints strings static int e3ga setFPPrecision const char prec set the string delimiters for all future prints strings static int e3ga setStringDelimiters char start char end The first function takes as argument an ASCIIZ string
21. a e1 and e3ga e2 are of the internal class type e3gai That s why gaigenhl h and gaigenhl cpp contain a combination of automatic type casting operator and extra inline defintions for some and operators func tions such that you can write the statements given above The definition of the casting operator looks like this inline e3gai operator e3ga amp const return e3ga this The operator casts variables of type e3gai to e3ga Examples of extra functions for using floating point variables and internal class variable in combination with multivectors are the following e3ga amp gp float a const e3ga amp b e3ga amp gp const e3gai amp a float b These function computes the geometric product of a float and an e3ga variable and of an e3gai and a float Such extra definitions are present in gaigenhl h and gaigenhl cpp for all functions where appropriate 3 19 Temporary Variables One may wonder where the temporary multivector variables come from in code like a b c d both lines are equivalent a add add b c d both line do the same thing since the program should first compute b c store the answer to that some where and then compute b c d possibly store that somewhere and then assign the result to a Where the temporary variables come from depends on whether you selected the fast temporary variable If you turned the fast temporary variables off the temporary variables are handled in t
22. a binary word Each bit which is on signals the de activation of a specific generate option Which options corresponds to chich bit can be looked up in gaspec h in the gaGenerateOption enumeration memfactor The single floating point argument of the memfactor key word is the factor entered in the storage tab It is only used when the memory allocation algorithm is set to balanced It must have a value of 1 0 or higher memman The memory allocation or management algorithm is specified by the string argument of the memman keyword The string argument can be one of tight balanced maxpp or max These values correspond to each of the four choices which can be made in the storage tab name The name of the algebra follows the name keyword optimize The optimize keyword specifies the optimization of a specific multivector product combination The syntax is optimize product gradeUsagel gradeUsage2 usage where product can be any of the abbreviated product names e g gp or Icont and gradeUsagel and gradeUsage2 are grade usage of each of the multivectors involved The grade usage should be interpreted as a binary word just like they are in Gaigen internally E g when bit 2 4 in decimal is on then grade 2 is in use The optional floating point value usage specifies how often this product was used according to a profile It is used by the ifelse and switch dispatching methods to ensure that the most often used function are found most ef
23. aigen user manual Available at http carol science uva nl fontijne gaigen 3 D Fontijne Gaigen Tutorial 1 Available at http carol science uva nl fontijne gaigen 4 S Mann L Dorst T Bouma The Making of GABLE A Geomet ric Algebra Learning Environment in Matlab in Geometric Alge bra with Applications in Science and Engineering E Corrochano and G Sobczyk eds Birkhauser 2001 pg 491 511 Also avail able at ftp cs archive uwaterloo ca cs archive CS 99 27 and http carol science uva nl leo clifford gable html 1999 5 T Bouma About the delta product Where ask Tim online 6 C Perwass The CLU and CLUDraw Library Available at http www perwass de cbup clu html 7 Bill Spitzak et al FLTK the Fast Light Tool Kit Available at http www fltk org 63
24. aration between the Gaigen program and the opt2X compiles was made to make it easy for 3rd party developers to gen erate low level computational functions for specific platforms they use The standard opt2X which come with Gaigen opt2c opt2c2 and opt2LAPack are called directly by Gaigen when it generates the source code and the opt file Which compiler is used can be controlled using the generate tab in the user interface opt cpp and opt h can be used to read and parse opt files Besides the function descriptions the opt files must contain three other key words e dimension the dimension of the algebra e classname the internal classname of the algebra e g e3gai e floattype the floating point type to be used float or double A general function description starts with a line like this general e3gai_general gp gp 0 2 opt cpp and opt h are also able to extract certain extra information from the products called product patterns These are used by the opt2c2 compiler Product patterns are the basic building blocks of the products Only a limited number of product patterns exist for each algebra and all of them are used by the geometric product The other products are made of selections of the product patterns of the geometric products Thus by implementing the product patterns of the geometric product all products can be implemented by calling the function which computes each required product pattern This is how the opt2c2
25. ary variables is by explicitly keeping references to temporary variables like this e3ga amp b C or by writing expressiong which are so long that they use all 64 tempo rary variables at once The number of temporary variable is controlled by the line define MV_MAX TEMP 64 in gaigenhl h In any case if you suspect that a malfunction of your pro gram is caused by using fast temporary variables you can simply turn off the fast temporary variables button regenerate your algebra recompile your application and see if the malfunction disappears fast dual The fast dual function can compute the dual of a multivector with respect to the pseudoscalar of the algebra very quickly compared to the default dual function by simply shuffling and flipping the sign of the coordinates reciprocal frame The reciprocal frame function can compute the recip rocal frame of a set of vectors multivector type This function can compute the type blade versor or general of a multivector Memory The memory tab as shown in figure 2 3b is used to control the memory allo cation method and the floating point type used to store the coordinates The 16 CHAPTER 2 THE USER INTERFACE coordinates of multivectors are stored in arrays of floating variables Gaigen stores only coordinates of grade parts of which it knows that they are not equal to 0 To save the amount of memory used to store the coordinates it can allo cate just enough mem
26. c c e3ga optc2 c or e3ga_optlapack c You need exactly one of these optimized files or you ll run into link errors If you use the e3ga optlapack c file be sure to include the LA Pack library in your project Also you can click the print tables button which will cause e3ga txt and e3ga tex to be generated These text and TpX files will contain the multiplication tables for the products of your algebra These may be useful for educational purposes To learn more about these tables see 2 or 4 To save the specification of your algebra to a file such that you can reuse it later click the save algebra button Gaigen will then ask you for a location to store the specification This stores every property of your algebra you can control through the user interface in the file The file format used is plain text Mf you take a look inside gaigenhl cpp and gaigenhl h you will see that the classnames GAIM_CLASSNAME and CLASSNAME are used in these files CLASSNAME is defined as the name of your class e g e3ga and GAIM_CLASSNAME is defined as the internal name of your class e g e3gai GAIM stands for Geometric Algebra IMplementation and is used in Gaigen source code as a prefix to macros that have to do with the intermediate C layer section 4 and the selections made in the gaigenui program 2 2 SIGNATURE 9 and is described in section 6 To load the specification of an algebra click the load algebra button A nice place to save your spe
27. cide to reallocate So the larger the waste factor the more memory may be wasted but the less memory reallocations will occur However this memory allocation scheme doesn t work well in practice as shown in 1 e Maximum The maximum number of memory locations to store all 2 coordinates is allocated when a multivector variable is created Gaigen never has to reallocate memory e Maximum parity pure We call a multivector parity pure if it is either odd or even If the dimension of the algebra is larger than 0 only half of the 2d coordinates have to be allocated to store the coordinates of a parity pure multivector variable The user must guarantee that he will never create multivectors that are not parity pure or weird things can happen a crash or incorrect results You can switch between the float 32 bit precision and double 64 bit pre cision types for coordinate storage by toggling the radio buttons on the right side of the tab If you always use the type GAIM_FLOAT defined as either float or double in your application you can switch back and forth painlessly between floats or doubles So instead of writing float c 1 0 2 0 3 0 e3ga a a set GRADE1 c 2 7 GENERATE 17 general signature products order functions storage generate OUPA Gir Se 1 generate 2 code generate LAPack code generate penttanies toad aigenra save algebra eat Figure 2 4 The generate tab with setti
28. cifications is in the algebras directory which is the default location where the save algebra dialog starts 2 2 Signature Clicking the signature tab will raise its contents which depend on the dimen sion of the algebra Since the signature tab is used to modify the signature of the basis vectors its contents are empty if you selected 0 as the dimension of your algebra in the general tab In figure 2 1b you can see the signature tab for a 3 dimension geometric algebra First of all the signature tab can be used to change the names of the basis vectors to something appropriate for your algebra using the textfields on the left E g if you want to generate an algebra to work with the conformal model of euclidean space you could give a special name to the basis vector represent ing the point at the origin and the point at infinity Or you could call the basis vectors of a 3d euclidean geometric algebra x y and z or red green and blue if you like The main purpose of the signature tab is to set the signature of the basis vectors The signature of a basis vector is the value it squares to e g we can define e Ve 1 e e 1 or even e e 0 a null vector The signature of all basis vectors is 1 by default but this can be changed to 1 and to 0 using the dropdown box provided for each basis vector It is also possible to create pairs of reciprocal null basis vectors A pair of reciprocal null vectors is a pair of vector
29. compiler works 58 CHAPTER 6 FILE FORMATS The general keyword is used to identify the start of a general function descrip tion The next string is the name of the function to be generated e3gai_general_gp in this case The next string is the abbreviated name of the product gp Fi nally the integer indicates that the products belongs to group 0 or group 1 this information is used to build the product patterns Multiple groups are used when euclidean and non euclidean versions of the metric products are required The function which should be generated in response to the example above is in C syntax void e3gai_general_ gp const float a const float b float c a and b are arrays of pointers to arrays of floating point values which contain the coordinates of the input multivector variables E g al0 points to the scalar coordinate of the first input multivector and b 2 points to the 3 bivector coor dinates of the second input multivector When a grade part is not used in the multivector variables the pointer is set to NULL e points to a single array of floating point variables This is where the results of the computition go The lines following the start of the general function description all look something like this c 4 alo 0 b 21 0 a 1 0 b 3 0 What this line says is that at c 4 the compiler should store a 0 0 b 2 0 a 1 0 b 3 0 This could almost directly be cop
30. ctor into an array of pointers to arrays of coordinates for each grade void e3gai expand const float pa 4 const Expand 2 multivectors a and b into arrays of pointers to arrays of coordinates for each grade void e3gai expand const e3gai amp b float const pal4 float const pb 4 const The profiling functionality is implemented by three functions two of which have already been described in chapter 3 The third internal function is incre mentUsage This function is called by the product functions when profiling is enable It counts in the array usageCount how many times each combination of product and multivectors has been used This information is printed when you call printProfile The productNames are used for printing purposes static int e3gai resetProfile static int e3gai printProfile float threshold 2 0 int e3gai incrementUsage int product int gua int gub static int e3gai usageCount 5 16 16 static const char e3gai productNames 5 Two algorithms are available to compute the join int e3gai joinAlgl const e3gai amp a const e3gai amp b int ga int gb int gj int e3gai joinAlg2 const e3gai amp a const e3gai amp b int ga int gb int gj Algorithm 2 builds up candidates for the join by wedging vectors together algorithm 1 projects blades of the right grade onto on of the multivectors a or b to find a good candidate for the join The best candidate wit
31. d algebra Thus special keywords are used which are replaced by the appropriate strings when the sourcefiles are generated These keyword are set by Gaigen and stored in a lookup table which also resides in the codeTemplateContainer class Each codeblock in gaspectemplates txt has the following structure CODEBLOCK codeBlockName ENDCODEBLOCK Each code block starts with a CODEBLOCK statement followed by the name of the code block That name is used to identify it the codeTemplateContainer class The statement starts with a dollar sign and is surrounded by curly braces lifa keyword is used in gaspectemplates txt which has not been entered into the lookup table the codeTemplate class prints out the name of the keyword in the generated source file making it easy to find such mistakes 55 56 CHAPTER 6 FILE FORMATS Each code block ends with aENDCODEBLOCK statement Between these two statements the code resides Anything can be put between the two statements so not just parts of C code Code or text outside a codeblock is ignored and generates a warning when parsed by the codeTemplateContainer class In the code block you can use keywords which will be replaced by strings from the lookup table An example of such a keyword is FLOAT When the code is output by the codeTemplateContainer class this keyword is changed to either float or double depending on the floating point type specified in the storage tab Thes
32. de of a homogenous multivector and 1 if the is not homogenous maxGrade returns the the index of the highest non zero grade The norm of a multivector according to one definition can be computed using the norm_a function float e3gai norm_a const The function is called norm a for now because other norm definitions will also be implemented To normalize a multivector use 4 5 INTERNAL FUNCTIONS 47 void e3gai normalize const e3gai amp a int norm The norm argument is used to specify which definition of the norm you want to use currently the only valid value is 1 which uses the norm_a norm To project a multivector onto a blade or versor or to reject a multivector from a blade one of the following functions can be used int e3gai project const e3gai amp blade const e3gai amp a int e3gai projectOntoVersor const e3gai amp versor const e3gal ta int e3gai reject const e3gai amp blade const e3gai amp a To compute the meet and join of two blades use 1 1 int e3gai join const e3gai amp a const e3gai amp b int algorithm int e3gai meet const e3gai amp a const e3gai amp b int algorithm The algorithm argument either 1 or 2 is used to specify which of the two available join algorithms to use The meet is computed with respect to the join These are three support functions used by the meet and join int e3gai deltaProduct const e3gai amp a const e3gai amp b int e3gai factor e3gai
33. e are often used keywords e CLASSNAME the name of the class as it appears in the high level C interface INTERNAL_CLASSNAME the name of the class used in the low level C glue FLOAT the floating point type float or double FLOATSIZE the size of the floating point type in bytes e g sizeof float FLOATCAST used to cast a type to a floating type e g double FLOAT_EPSILON the value that is consider small by default le 8 for float and 1e 15 for double e NUMBER_OF_COORDINATES the total number of coordinates 2 DIMENSION the dimension of the algebra e g 3 DIMENSION 1 the dimension of the algebra plus 1 1 lt lt DIMENSION 1 2 to the power of the dimension plus 1 2 MAX_NUMBER_OF_COORDINATES only used when the memory allocation algorithm is max or maxpp Is equal to the maximum number of coordinates which can be stored in the coordinate array MEMFACTOR the balaned memory allocation algorithms waste fac tor NUMBER_OF_PRODUCTS how many products are included in the algebra includes euclidean versions of the geometric and scalar product and left contraction GP_EM the name of the function which can compute the euclidean metric geometric product used for meet and join LCONT_EM the name of the function which can compute the eu clidean metric left contraction used for meet and join SCP_EM the name of the function which can co
34. e static reciprocal frame function computes the re ciprocal frame of a set of basis vectors Its definition is static int reciprocalFrame c3gai f const c3gai ell int nbVectors This sets the set of nb Vectors vectors f to the reciprocal frame of e 3 16 Profiling To use the profiler the profile checkbox in the function must be ticked Gaigen will then include profiling code when the algebra is generated Only two static functions accessible by your application are added resetProfile and printPro file Static means that the functions are accessible without reference to an in stantiation of the class The following e3ga printProfile will invoke the static printProfile function in the class e3ga Thus when you use a static function of a class you don t have to have access to a variable of that class Actually the three profiling functions are always present in the algebra code whether the checkbox is ticked or not But when it is not ticked the generated function does nothing If the functions would not be included at all in the source code you would have to remove or add your calls to the profiler each time you turn the profiling option off or on The profiling code slows down the performance of your application so its best to turn it off when you don t use it anyway resetProfile resets the count of all product multivector combinations You should call this function at the start of your application printProfile
35. ed they are indices in the range 0 d 1 The signature can be either 1 or 1 The signature 0 is forbidden since then you could just as well create two ordinary null vectors signature The signature keyword specifies the signature of a basis vector The first argument is the index of the basis vector range 0 d 1 and the second argument is either 1 0 or 1 gap files gap files are written by the saveProfile function There are four possible key words in a gap file profiledate The profiledate keyword is followed by a string specifying the date and time when the file was written name The argument to the name keyword specifies the name of the algebra dirname The argument to the dirname keyword specifies the directory name where the algebra was generated Both name and dirname are used to identify the algebra i e to make sure you don t use the wrong argument productusage The productusage specifies what product of what types of multivectors is used how often These lines form the bulk of a gap file The format of the keyword arguments is identical to those of the optimize keyword in a gap file productusage product gradeUsagel gradeUsage2 usage 62 CHAPTER 6 FILE FORMATS Bibliography 0 To do fix some of these references 1 D Fontijne T Bouma L Dorst Gaigen A Geometric Algebra Implementation Generator Available at http carol science uva nl fontijne gaigen 2 D Fontijne G
36. eous functions a 37 3315 1 normal ia dt da a a A T 37 3152 dual A An 9 37 915 3 MV ALY Per tao pa e e a a 37 3 15 4 layer 1 functions o ooo ooo 38 316 Profilng ss cia a altas Ble Ba de lt ida 38 CONTENTS 3 17 Basis Vectors and Inverse Pseudoscalar 39 3 18 Use of Internal Class and Floating Point Variables in Functions 40 3 19 Temporary Variables 2 0 0000000000008 40 3 20 Multitheading 02 624 eta reirse iese aatia 42 ZA MIEAWIDS san Sek a A kt aby ce Gece a Aid 42 Layer 1 Internal C class 43 AV CH Class ela e eke gerd te eae a e 43 4 2 Constructors Assignment Functions 45 ta hermdi ds ee ee E A hes oh 45 4 3 1 Euclidean Metric Products 45 44 Other Functions o e e ee eee 46 4 5 Internal Functi0NS o o e o 47 4 6 Global Internal Variables ooo 48 Layer 0 low level computational functions 51 File Formats 55 6 1 gaspectemplates txtfile o o o o ooo oo oo 55 6 2 opt files ia a a a A AER 57 6 3 Castle a td E dea Be 59 64 gaptiles corra ara beh se Rael Saat 61 Chapter 1 Introduction Gaigen is a program that can generate implementations of geometric algebras It generates C C and assembly source code that implements a geometric al gebra requested by the user This is an unconventional approach The choice to create such a program package was made becau
37. esto inverse tast Qual De Al es general inverse e reciprocal frame substract spinor product mubvector type taxe grade catermorphesen highest grace e factor e race meet and jon genere pert tables load algebra save algebra et pert tables load algebra save algebra a b Figure 2 3 The functions and memory tabs with settings for the e3ga algebra otherwise you might run into compilation errors If you try to compile an algebra and the compiler complains about a certain function or product which does not exist include it from the functions or products tabs Because the names of most functions speak for themselves what follows is a list of a functions with peculiarities or names that do not make 100 percent clear what the function does How to use the functions is explained in section 3 e take grade This function takes one grade part from a multivector and copies it to a new multivector e highest grade This function takes the highest grade part that is non zero from a multivector and copies it to a new multivector It is used to compute the delta product e grade of a blade When given a blade or a homogenous multivector this function returns its grade If a non homogenous multivector which is is passed the function returns an error e norm We have implemented several currently two functions to com pute the norm of multivectors However every application seems to have
38. evel C or as sembly code implements the actual computation of products of the geometric algebra and is described in chapter 5 This chapter also describes how one could implement his or her own version of this code optimized for a specific processor architecture Finally chapter 6 describes the various file formats used by Gaigen Chapter 2 The User Interface It may seem a bit weird to download a programming package and the first thing to do with it is start its user interface but that is exactly how Gaigen works unless you use one of pre generated sample implementions that come with Gaigen The user interface that pops up after starting the gaigenui exe cutable is used to specify the properties e g the dimension of the algebra sig nature of the basis vectors optimizations of the geometric algebra you would like to use After selecting those properties you hit the generate button and source code for your algebra is generated Then you can exit the Gaigen pro gram and are ready to use your algebra You don t have to use the Gaigen user interface again unless you would want to change properties of your algebra or perhaps generate source code for an entirely new one or if you would like to change the performance optimizations of your algebra Changing the optimizations of the algebra can increase the performance of your program drastically Gaigen can include profiling code in your algebra that tracks what products multivector
39. f code is very de pendent on the orientation and order of the grade 2 basis blades If you change the orientation and order of these blades using the order tab see section 2 4 this code would function incorrectly A better way to retrieve individual coordinates is using the EZGA_X macros defined in e3ga h For each coordinate a macro is generated which specifies the index of the coordinate in the coordinates array relative to the first coordinate of that grade part For the e3ga algebra these macros look like this define E3GA S 0 define E3GA I 0 define E3GA El 0 define E3GA E2 1 define E3GA E3 2 define E3GA E2 E3 0 define E3GA_E3 El 1 define E3GA El E2 2 If you change the order of the coordinates these macros automaticly change as well when you regenerate the code It s easy and readable to use the macros like this to retrieve the individual coordinates get each of the grade 2 coordinates of a float g2c a coordinates GRADE2 float g2c_el e2 g2c E3GA El E2 float g2c_e2 e3 g2c E3GA E2 E3 float g2c e3 el g2c E3GA_E3 El The only instance where you could run into trouble with accessing coordi nates like this is when you flip the orientation of the basis blades For instance this could change define E3GA E3 El 1 into define E3GA El E3 1 and then your application won t compile anymore if you still use E3GA_E3_E1 But this is better than compiling successfully and then failing at runtime be cause your a
40. f the algebra as specified by the dimension keyword dimension The single argument of the dimension keyword specifies the dimension of the algebra It should always be the first keyword in the gas file Only lines with comment or blanks can preceed it dirname The string argument to the dirname keyword specifies the di rectory relative to the algebras directory where the generated code will be stored This is equal to the name of the algebra by default but this be changed in the generate tab floattype The floattype keyword is used to set the type of floating point numbers floats or doubles which the generated source code uses Gaigen is not yet fully able to generate source code which uses doubles so the user interface does allow the user to set the the type of floating point numbers yet function The function keyword has a single argument which is an inte ger The value of the integer specifies which functions are included in the 60 CHAPTER 6 FILE FORMATS algebra These functions correspond to the checkboxes in the function tab and the gaFunction enumeration in gaspec h The integer should be interpreted as a binary word Each bit which is on signals the inclusion of specific function in the algebra Which function corresponds to which bit can be looked up in gaspec h in the gaFunction enumeration generateoption Like the function keyword the generateoption keyword is followed by an integer which should be interpreted as
41. factors int versor 0 const int e3gai factorVersor e3gai factors const The coordinates function returns a pointer to the coordinates of a certain grade part You should not try to modify the floating point variables this pointer points to const float e3gai coordinates int grade const To compute the reciprocal frame of a set of nbVectors vectors e use recip rocalFrame The result goes into f static int e3gai reciprocalFrame e3gai f const e3gai ell int nbVectors The outermorphism has the same construction and initialization functions as the high level C interface outermorphism except that there is no init with spinor initializer The print and string functions are exactly those described in the high level C interface The fastDual function can be used to compute the dual with respect to the entire space int e3gai fastDual const e3gai amp a 4 5 Internal Functions This section described several interesting internal functions The setUsage function sets the grade part and memory usage of a multi vector variable It makes sure enough memory has been allocated to store the coordinates It s declaration is void e3gai setUsage int u 48 CHAPTER 4 LAYER 1 INTERNAL C CLASS These are several functions to expand the coordinates of a multivector Expand a multivector into a matrix according to table void e3gai expand float matrix const int table const Expand a multive
42. ficiently order The order keyword specifies for each index in the uncompressed coordinate array from 0 to 21 1 what basis blade coordinate is stored there The basis blades are specified by which basis vectors are included in them The basis blades are again given as a binary word where each bit signals the in or exclusion of a basis vector So the line order 7 6 tells us that at uncompressed coordinate array location 7 the coordinate relative to the basis blade e2 ez is given If an optimization is specified for a product which is not included in the algebra it is still read and stored by gaigenui but it won t be visible in the products tab until the product is included 6 4 GAP FILES 61 6 4 product The product keyword is followed by any of the any of the abbre viated product names e g gp or Icont and specifies the inclusion of that product in the algebra sign The sign keyword is followed by a basis blade specifed as a binary word and either 1 or 1 It specifies the orientation of the basisblade When you to toggle the orientation for a basis blade in the order tab you toggle reciprocal The reciprocal keyword is used to specify reciprocal null ba sis vectors In the user interface this is done by checking the reciprocal checkbutton between two basis vectors in the signature tab The recip rocal keyword is used like this reciprocal 3 4 signature 1 The 3 and 4 indicate which basis vectors are involv
43. flavours just like the constructors above The first flavour can only be used to set a multivector variable to a homogenous value a set GRADE1 1 0 2 0 3 0 This sets the existing multivector variables a to le 2e2 3e3 To set the value to a non homogenous value the other flavour of the set function can be used float coordinates 8 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 a set GRADEO GRADE1 GRADE2 GRADE3 coordinates This example would assign the value 1 2e 3e2 4e3 5e2 e3 6e3 Ne Tei A eo 8e A eg A e3 to a A variation of the set function are the named set functions a setScalar 1 0 b setVector 2 0 3 0 4 0 c set3Vector 8 0 float coordinates 3 4 0 5 0 6 0 d set2Vector coordinates These can take both single coordinates and coordinates provided in arrays and are available for all grades of the algebra A final variation of the set functions is the use of the operator to assign a scalar value to a multivector variable 24 CHAPTER 3 LAYER 2 HIGH LEVEL C INTERFACE ALOE Note that the use of the is made possible through operator overloading To copy the value of one multivector variable to another use a b or a copy b which would copy the value of b into a To set a multivector variable to a random value the random blade versor function can be used The following example sets the multivector variable a to a random scalar a randomBlade GRADE0 1
44. generate general signature products order funcnons storage generate gp nip manip cont rcont op scp optimacations 909192193 Geometric Prouet toggle e2e3 own opbmize wage Jere w com 0123 0123 6696 6556 aid ad 0 2 1 remove 1 3 0 2 remove generate perttanies toad aigenra save algebra ext generate penttanies toad aigenra save algebra et Figure 2 2 The products and order tabs with settings for the e3ga algebra In Gaigen all multivectors are stored as compressed arrays of coordinates A multivector A 1 2e 3e1 4e 5e1 A e2 Ge A e3 Tez A e3 8e1 A e2 A e3 2 2 would by default be stored as an array 1 2 3 4 5 6 7 8 But suppose you don t like the orientation of the basis blade e ez and prefer ez e instead You could do this by clicking the g 2 tab g 2 stands for grade 2 and click ing the toggle button for e A ez which will then change into ez A es The multivector A with the same value as above would then have to be stored as 1 2 3 4 5 6 7 8 Toggle buttons are active only for basis blades with a grade higher than 1 Now suppose you want to change the order in which the coordinates are stored in the array e g because the order in which you store your coordinate data is different You would use the up and down buttons to change the or der of basis blades Note that you can only modify the order of basis blades within a grade All coordinates for one g
45. h the largest norm is returned as the join Both algorithms take as arguments two multivectors a and b the grade of each multivector ga gb and the required grade of the join gj The inline functions gradeUsage and memUsage return the grade part and memory usage of a multivector This information is extracted from the usage variable in the multivector inline int gradeUsage const inline int memUsage const 4 6 Global Internal Variables There are a number of interesting variables outside the C which are assumed to be useful to the internal class only that s why they are not included in the class itself To name a few 4 6 GLOBAL INTERNAL VARIABLES 49 const char e3gai_basisElementNames 8 int e3gai_gradeSize 4 int e3gai_mvSize 16 e3gai_basisElementNames is an array of strings which holds the names of all the basis vectors e g e1 e2 and e3 These names are used for printing the coordinates of a multivector variable The integer array e3gai_ gradeSize con tains the number of coordinates each grade part holds e3gai mvSize is also an integer array and holds for each possible combination of grade usage the minimum number of coordinates required to store the coordinates This is used among others for the memory allocation algorithms The grade part usage which serves as an index into the e3gai_mvSize array is specified in the same binary format as in figure 4 1 50 CHAPTER 4 LAYER 1 INTERNAL C CLA
46. he default C way All functions and operators like add and operator return plain multivector variables like this 3Note that because of this casting operator you can not add non static member variables or virtual functions to gaigenhl h and gaigenhl cpp Doing so anyway might mess up the simple but naive casting operator 3 19 TEMPORARY VARIABLES 41 e3ga e3ga operator const e3ga ta e3ga add const e3ga amp a const e3ga b Note the absence of the amp symbol in the return type This means that an actual variable is returned instead of just a reference These variables are allocated from the stack and then initialized using the standard constructor the code to do so is generated by the C compiler One can imagine that allocating and initializing a multivector variable for each intermediate result can lower the performance of your application On the other hand if you turned on the fast temporary variables functions and operators returning multivector variables will be defined like this e3ga amp e3ga operator const e3ga ta e3ga amp add const e3ga amp a const e3ga b The functions and operators now return references to multivector variables note the amp symbol in the return type The actual temporary variables are allocated by Gaigen source code from an array of multivectors variables with a fixed size This is more efficient than letting the C compiler allocate them from the stack but it has a dra
47. hence the amp and are good operator symbols for them The join is on based the delta product 5 and algorithms described in 1 The delta product is used to compute the grade of the join of the two input blades Then one blade is factored into a number of vectors which are then repeatedly wedged to the other blade until a blade best representing the join of the input blades is found The meet is compute using the join as described in 1 The following shows how to use the meet and join Assuming b and c are two blade valued multivector variables this code computes their meet and join e3ga m b c e3ga j b amp C which is equivalent to e3ga m meet b c e3ga j join b c The definitions of the meet and join functions are e3ga amp meet const e3ga amp a const e3ga amp b e3ga amp join const e3ga amp a const e3ga amp b e3ga amp e3ga operator amp const e3ga amp a const e3ga amp e3ga operator const e3ga amp a const 3 10 Factorization The function factor and factorVersor can be used to factor blade or versors into vectors When the vectors are wedged or multiplied together they rebuild the blade of versor The factor function is used by the meet and join functions and a use of the factorVersor function might be to retrieve the vectors in the plane of rotation of a rotor The definitions of the factorization functions are 3 11 EXPONENTIATION 31 Factor a blade versor
48. id e3gai_addSameGradeUsage float c 51 52 CHAPTER 5 LAYER 0 LOW LEVEL COMPUTATIONAL FUNCTIONS a 0 scalar b 0 scalar L1 e1 1 NULL e2 e3 2 e2 e3 2 e2 e3 e3 e1 e3 e1 e1 e2 e1 e2 L3 pseudoscalar 3 NULL Figure 5 1 Array representation of multivector variables a and b const float a const float b int length add two multivectors with the different grade part usage foal lal DO void e3gai_add const float a 4 const float b 4 float c 4 subtraction function simular to addition functions void e3gai_subSameGradeUsage float c const float a const float b int length void e3gai_sub const float a 4 const float b 4 float c 4 All these functions must always be implemented by the code generated by the opt2X compiler As you can see all functions except e3gai_copy e3gai norm a e3gai_negate e3gai_addSameGradeUsage and e3gai subSameGradeUsage take arrays of pointers to arrays of floating point variables as arguments These ar rays contain the coordinates of each grade part A pointer to a coordinate array can also be NULL when a grade part is not in occupied by the multivector This is illustrated in figure 5 1 It shows the array representation of multivectors a and b a occupies all grade parts so all 4 pointers actually point to coordi nates b on the other hand is a rotor and only occupies grade part 0 and 2 The pointers for grade part 1 and
49. ied and pasted to make up a valid C function except that one should check for NULL pointers in arrays a and b The optimized function descriptions have a slightly different syntax than the general functions Here is an example of an optimized function description which is to compute the geometric product of a rotor and a vector optimize e3gai_opt_05 gp 02 gp 0 05 02 c 0 aro b 0 a 3 b 1 a 2 b 2 c 1 a 3 b 0 a o b 1 a 1 b 2 c 2 al2 b 0l al1 b 1 alo b 2 c 3 al1 b ol a 2 b 1 a 3 b 2 Again the first keyword optimize is used to identify the start of the optimized function description This is followed by the name of the function tha abbriv iated name of the product the group number 0 and the grade usage of the input multivectors in hexadecimal format 05 and 02 in this case opt2c gener ates this function is response to the functino description above void e3gai_opt_05 gp 02 const float a const float b float c crol alo b 0 al3 b 1 al2 bl2 c 1 al3 b 0 a 0 b 1 all bl2 6 3 GAS FILES 59 c 2 a 2 b 0 a l b 1 a 0 b 2 c 3 a l b 0l a l2 b 1 al3 b 2 As you can see compiling optimized functions is even simpler than compiling general functions Besides the general and optimized functions the opt2X compilers also gen erate a number of low level computati
50. internals look like e an installation manual e a paper describing and discussing the design of Gaigen and reporting its performance relative to other packages and itself using different set tings e the Gaigen executable for win32 most flavours of Windows Sun Solaris and Linux and its source code 6 CHAPTER 1 INTRODUCTION e pre generated algebras for euclidean e3ga projective p3ga and con formal model for 3d geometry c3ga e tutorials showing how to use the pre generated algebras and how to gen erate and use your own algebras and e a quick reference page for the high level C interface This user manual is divided into six chapters of which most users not in terested in modifying or changing Gaigen will only have to read the first two to get started These chapters are chapter 2 that describes the Gaigen user inter face and chapter 3 which describes the high level C programming interface of the source code Gaigen generates The source code generated by Gaigen is exposed to the user as a C class with member functions and overloaded operators No fancy C features except for operator overloading are used to keep everything as basic and simple as possible Gaigen does not depend on other software packages except for its user interface which uses the FLTK library 7 Chapter 4 describes the intermediate C layer that lies between the low level C or assembly code and the high level interface The low l
51. ion to make sense the grade of the object on the lhs should be smaller than the grade of the object on the right hand side right c a gt gt b Same reasoning as with the left con oe contraction a gt gt 7 traction scalar The symbol is used for modulo di product b vision in C the scalar product kind of works like a modulo returning only the scalar part of a geometric product To remember the symbol imagine the two circles in the symbol being two zeros indicating that only the grade 0 part will be returned addition c a b a b subtraction c a h negation 3 1 CONSTRUCTION 21 inversion lb The symbol is used to denote the bi nary complement of integers in C This suggests its use as the inversion operator for multivector since the in verse of a multivector could be consid ered its complement reversion a b The superscript symbol is often used a b in geometric algebra literature to de note the reverse This operator can be used both pre conjugate and postfix Both do the same and do not change the operand unlike the operator when applied to standard C integers of floats grade a b Same comment as the operator involution amp amp meet The amp symbol is used for the binary and operation for integers in C The meet is like an and operation for sub spaces a c a b The symbol is used for the binary or a b operation fo
52. iprocal ele2 name Eo e2 ee 1 reciprocal eled high level C interface a ul par CESE generate penttaies toad agenra save aigetea ea generate penttanies toa aigenra save algebra eat a b Figure 2 1 The general and signature tabs with settings for the e3ga algebra by Gaigen to implement the products becomes infeasible for higher dimension see 1 Generating a 0 dimensional algebra will basically give you a scalar algebra If you want to use the high level C interface described in section 3 make sure the checkbox high level C interface is ticked Gaigen will include two files gaigenhl h and gaigenhl cpp in the source code for your algebra The name of your algebra can be entered at the name textfield This name is used to generate the output filenames and as the name of the class which is generated Also the name is used internally with an i appended to it as the name of the class which sits between the low level code and the high level C interface In the discussion and figures below we assume that you call your algebra e3ga When you click the generate button at the lower right the source code and other files will be generated in your algebras directory which you set in your configuration file see the installation manual The files you have to include into your programming project are e3ga cpp e3ga h and depending which checkboxes you activated in the generate tab either e3ga opt
53. its own idea of what the of a multivector norm is so these functions have not entirely stabilized yet e normalize Normalizes a multivector by dividing it by its magnitude norm Uses the outer product and the norm e versor inverse Computes the inverse of a multivector assuming it is a versor a versoris a multivector that can be written as the geometric prod uct of vectors If the multivector is not a versor something other than the inverse is returned The versor inverse is very efficient and should al ways be preferred over the general inverse function if possible Requires the reverse scalar product and the outer product This should never happen but we still have to build a good dependency system that checks that all required products and functions are present 14 CHAPTER 2 THE USER INTERFACE e lounesto inverse This function can compute the inverse of any invertible multivector in a 3 dimensional algebra Requires the clifford conjugate and take grade functions and the geometric product See 4 and 1 for more information on how it works general inverse This function can compute the inverse of any invertible multivector in an algebra of any dimension It computes the inverse by explicitly inverting the geometric product matrix expansion It is slow compared to the versor inverse or the lounesto inverse See Gaigen tuto rial 1 3 for an example of relative efficiency and 1 for details Does not require any
54. l its variations e3ga amp e3ga operator const e3ga amp a const e3ga amp operator float a const e3ga amp b e3ga amp e3ga operator float a const e3ga amp e3ga operator const e3ga ta e3ga amp e3ga operator float a the three inverse functions e8ga amp e3ga versorInverse const e3ga amp e3ga lounestoInverse const e3gas e3ga generalInverse const the functions which do the same as the operators igp stands for inverse geomtric product e3ga amp igp const e3ga amp a const e3ga amp b e3gas igp const e3ga amp a float b e3gas igp float a const e3ga amp b Remember that every time you use the inverse geometric product or the operator you are implictly inverting a multivector So if you have to divide by the same multivector value many times it is more efficient to first invert the multivector store that inverse and then multiply by the inverse instead of dividing by the original multivector 3 6 Addition subtraction negation Multivector variable can be added and subtracted using the add sub and functions and operators The negate operator is also used to compute the negation of a multivector variable e3ga a b C a b this is equivalent to a b negate this which is equivalent to a 0 0 b this The definitions of these the addition subtraction and negation functions are addition e3ga amp e3ga operato
55. liffordConjugate const grade involution e3ga amp e3ga operator const e3ga amp e3ga operator int const e3ga amp e3ga gradeInvolution const 3 8 Grade Part Selection The grade function and the operator can be used to select a certain grade part from a multivector variable This may be useful when you are only interested in a certain grade part and not in any others such as in this example e3ga vector rotor vector rotor vector rotor inverse vector vector GRADE1 30 CHAPTER 3 LAYER 2 HIGH LEVEL C INTERFACE Here some vector is rotated by some rotor We know that the vector is a grade 1 blade and should still be a grade 1 blade after the rotation However due to floating point round off errors a small grade 3 part may arise This grade 3 part is thrown away by the the third line in the example the GRADE operator call selects only the grade 1 parts of vector This example could have been one line shorter like this e3ga vector rotor vector rotor vector rotor inverse GRADE1 The definitions of the grade function and the operator are e3ga amp e3ga grade int g const e3ga amp e3ga operator int g const 3 9 Meet and Join The amp and symbols are used as operators for the meet and join of blades or subspaces These symbols are used as the binary and and or operations when used on integers The meet and join compute the and and or of subspaces
56. matrix representation for the grade 1 part is exactly the traditional matrix that would be used when one would use linear algebra to do geometry The 1x1 matrix that transforms the pseudoscalar is the deter minant of the transformation 1 The matrix representation leads to another advantage of the outermorphism because of the matrix form it can easily be executed using Single Instruc tion Multiple Data SIMD instructions sets such as supplied by the SSE 2 3DNow or AltiVec This could be exploited by a future opt2X compiler The definitions of the outermorphism constructors are construct but don t initialize e3ga_om e3ga_om construct amp initialize using array of vector images e3ga_om e3ga_om const e3ga vectorImages 3 construct amp initialize using array of pointer to vector images e3ga_om e3ga_om const e3ga vectorImages 3 construct amp initialize using spinor rotor versor e3ga_om e3ga_om const e3ga amp spinor The initialization functions are init using a spinor rotor versor to create the vector images int e3ga_om initSpinor const CLASSNAME amp spinor init using images under the outermorphism of the basis vectors 1The other grade parts of the matrix represention are not widely know traditionally but are also very useful 3 13 COORDINATE OUTPUT AND ACCESS 33 int e3ga_om initVectorImages const CLASSNAME vectorImages 3 int e3ga_om initVectorImages c
57. mbol so some will have to do with regular function names Functions are available for all operations and the code above could be written as a copy gp gp R b R inverse which gives the same result but isn t half as readable We have done our best to make a reasonable operator selection for the operators and also provided the 19 20 CHAPTER 3 LAYER 2 HIGH LEVEL C INTERFACE reasoning behind each operator but there will always be people who disagree with our choice Already our selection differs at some points from the selection made in another package CLU 6 We now list the functions assigned to all overloaded operators with a com ment as to why we assigned that specific function to that specific operator We will discuss the use of operators later on this is only a reference table operator function sample comment assignment as geometric The symbol is used for multiplication product in C Thus it is the best for the most fundamental product in geometric al gebra division inverse geometric product outer c anb Best visual match to wegde symbol product a b lt lt is the binary shifting operator in contraction a lt lt A C The left contraction could be thought of as shifting or removing the lhs argument from the right hand side argument A trick to remember the symbol the lt symbol is used to express the smaller than relation in mathematics For the left contract
58. mpute the euclidean metric scalar product used for meet and join VERSORINVERSE EM the name of the function which can compute the euclidean metric versor inverse used for meet and join 6 2 OPT FILES 57 e PROJECT_EM the name of the function which can compute the eu clidean metric projection used for meet and join e TMPVAR1 temporary variable set right before the code block is out put The value of these temporary variable varies from code block to code block e TMPVAR2 temporary variable e TMPVAR3 temporary variable 6 2 opt files opt files are genereted by Gaigen and contain descriptions of the low level computational functions which must be implemented to compute the prod ucts There are two types of low level computational functions general and optimized The general functions must always be implemented for every prod uct that is included in the algebra they must are able to compute the product of any multivector with any other multivector independent of grade usage Descriptions of optimized functions however are generated for every opti mization that was entered in the products tab Optimized function can only compute the product of two specific multivectors e g the outer product of a vector and a bivector The opt files are read by special opt2X compilers which compile them into source code e g C C or assembly The job of the compilers is to do this as efficient as possible The sep
59. nd assign multivector or float void e3ga copy const e3ga ta e3ga amp e3ga operator const e3ga amp a e3ga amp e3ga operator float f set to random blade or versor int e3ga randomBlade int grade float scale int e3ga randomVersor int grade float scale int e3ga random int grade float scale int versor 3 3 Operators As already discussed in at the start of this section a number of often used op erations such as addition geometric product and outer product have special symbols assigned to them i e x and A This is done through a C fea ture called operator overloading which allows the programmer to assign new meaning to a symbol when itis used in combination with a class It drastically increases the readability of code e g compare a R b R inverse with a copy gp gp R b R inverse The former using operator overloading is much easier understood though the C compiler will generate the same machine code for both cases One could classify the operators into thee types binary and unary and in place Binary operators such as take two arguments e g a beak oe which is equal to a add b c On the contrary unary operators e g take one argument e g a b which is equal to a b reverse j And last of all the in place operators take two arguments one of which is also used to store the result e g 26 CHAPTER 3 LAYER 2 HIGH LEVEL C
60. nge if you wanted to for instance if you don t like the operator definitions This chapter discusses not only the features provided by gaigenhl h and gaigenhl cpp but also some of the more convenient features provide by the lower level C interface discussed in chapter 4 because those features are di rectly accessible In short we could say that this chapter discusses everything the casual user needs to know about programming geometric algebra using source code generated by Gaigen One of the most important features introduced by the high level interface is operator overloading Operator overloading allows a programmer to give new meaning to a symbol e g and to almost whatever he or she wants This allows us to write things like a R b IR to express a RbR This example immediately shows a problem with op erator overloading picking the right operator symbol for the right operation The symbol is used for the scalar product in geometric algebra literature but is used for general multiplication in C thus one could reason it is a good choice for the geometric product The symbol is used in C arithmetic as the not or binary complement operator for integers which might make it a good choice for the inverse but also for the dual As you can see it is not easy to pick the right symbol for each operation Furthermore there are not enough operator symbols available to give every operation its own sy
61. ngs for the e3ga algebra write GAIM_ FLOAT lis 2 0 3 0 e3ga a a set GRADE1 c When you switch floating point type the type of the array of coodinates c is automatically switched too If you use multiple algebras in one application make sure they use the same floating point type or you might run into trouble This might be improved in the future As shown with benchmarks in 1 the combination of maximum parity pure memory allocation and the use of floats as floating point type leads to the high est performance For high dimensional algebras e g 5D and doubles the tight memory allocation method might be more efficient however 2 7 Generate The generate tab shown in figure 2 4 is used to control several aspects of the code generation The output dir textfield has the same contents as the name field in the gen eral tab by default when you change the name field the output dir field is set to the same contents as the name field However you may find it useful to have several algebras with the same name e g e3ga but with different prop erties such as optimizations products and functions One instance where this may be useful is when you use the same algebra in different applications Different applications may benefit from different product optimizations but instead of using two algebras with the different names which would become bothersome when you use code from one application in the other you can use two
62. o the pseudoscalar I of the algebra e3ga a b El b a dual set b to the dual of a the dual is computed as b alI If the fastDual function is included in the algebra then that function is used by default It works by swapping and negating the coordinates which is faster than explicitly computing the dual 3 15 3 mvType mvType returns the type of a multivector GA BLADE GA_VERSOR or GA_MULTIVECTOR If the multivector is a blade it also returns the grade of the blade It s declaration is int mvType int grade double epsilon const The pointer to grade can be NULL if you don t care about the grade The value in grade is one of the constants GRADE0 GRADE GRADEN The value of epsilon defaults to 1e 14 for doubles 1e 7 for floats A usage example e3ga a GRADE1 1 0 2 0 3 0 int grade int type a mvType amp grade type GA BLADE grade GRADE1 2 38 CHAPTER 3 LAYER 2 HIGH LEVEL C INTERFACE 3 15 4 layer 1 functions There are also a number of functions from layer 1 that are useful and can be accessed directly from the high level C interface e null The null sets its argument to 0 e compress compresses its argument in place This means to remove any grade part that is equal to 0 You can supply an option floating point argument e to compress to tell it what is still considered to be 0 e3ga a PE wae a compress le 10 compress all grade parts lt le 10 reciprocalFrame Th
63. onal functions which can add subtract negate reverse etc multivectors These are to be described elsewhere 6 3 gas files gas files store Gaigen s specifications of geometric algebras gas stands for Geometric Algebra Specification The gas file format is very simple Every line consists of a keyword e g dimension and several arguments Keywords can appear in any order with one exception the dimension keyword is always the first keyword of the file Comments begin was the hash symbol and end at the end of a line Keywords are case insensitive and if they are unknown to the parser the entire line is ignored If you look in the gaspec cpp source code functions that have to do with gas files have names like loadProfile and saveProfile Initially the word profile instead of specification was used to refer to the properties of an Gaigen algebra The word profile was not exposed to the user later on because it may cause confusion in relation with the profile checkbutton and function This is a list of all keywords arguments and descriptions in alphabetic or der basiselementname The basiselementname keyword has two arguments The first argument is a number followed by a colon e g 0 the second a word which is a valid C variable name e g e1 The basiselement name keyword is used to set the name of the basis elements or vectors of the algebra Basis elements are labeled from 0 to d 1 where d is the dimension o
64. onst CLASSNAME vectorImages 3 To apply the outermorphism to a multivector variable use either of these functions apply outermorphism L to multivector A returns the result e3ga amp om const e3ga_om amp L const e3ga amp A the operator does the same as the om function e3ga amp operator const e3ga_om amp L const e3ga amp A 3 13 Coordinate Output and Access You may want access to coordinate for several reasons One reason is that you may want to store your multivectors in files to retrieve them later This can be done by storing their coordinates and later restoring them when your application reads the file Another reason is inspection of the coordinates either graphically or by printing them as numbers Although geometric algebra is coordinate free and Gaigen is as well after you have entered the coordinates into multivectors most people find it useful at least while still somewhat uncomfortable with geometric algebra to inspect the coordinates of multivectors We will first treat printing the coordinates via Gaigen and then show how your application can get get access the coordinates If you want to print the coordinates to the standard output you can use the print function like this e3ga a GRADE1 1 0 2 0 3 0 a print a This example would print a 1 00 el 2 00 e2 3 00 e3 The default way the floating point coordinates are printed is using the printf format 2 2f
65. or inverse is not available the lounesto inverse is used and finally it resort to the general inverse This is handled by the following piece of code in gaigenhl h ifdef GAIM FUNCTION VERSORINVERSE inline void inverse const CLASSNAME amp a versorInverse a elif defined GAIM FUNCTION LOUNESTOINVERSE inline void inverse const CLASSNAME amp a lounestoInverse a elif defined GAIM FUNCTION GENERALINVERSE inline void inverse const CLASSNAME amp a generalInverse a endif This automatic selection of the inverse function shouldn t problem be a prob lem unless you want to force the use of specific inversion function One case where this might occur is if you want to write a b c inverse where c is not a versor If you have included the versor inverse Gaigen will apply it even though it will give the incorrect answer To force Gaigen to use a specific inversion function you have to explictly name the inversion function you want a b c versoriInverse a b c lounestoInverse a b c generaliInverse 28 CHAPTER 3 LAYER 2 HIGH LEVEL C INTERFACE This of course rules out the use of the inverse function and the operator because they always resort to the default inversion function The definitions of the inversion functions are compute the inverse using the default algorithm e3ga amp e3ga inverse const e3ga amp e3ga operator const the overloaded operator in al
66. or the geometric product between an even multivector and a vector Rv and an optimized function for the geometric product between an odd multivector and an even multivector Rv R Because Gaigen tracks the grade usage which grade parts of a multivector are equal to 0 empty and which are not of all multivectors you use in your ap plication it is in theory capable to generate and invoke an optimized function for every combination of grade usages and products Of course generating an optimized function for every possible combination is not feasible for high di mensional algebras because the amount of code generated would get too large That s why you can select a specific set of combinations from the products tab You turn on optimization for a specific combination of grade usage and product by first swiching to that product click its tab Then you use the two sets of little checkboxes labeled 0 d where d is the dimension you selected for your algebra in the general tab to specify the grade usage combination you want to optimize for and add it to the set of optimizations by clicking add So suppose you want to optimize for the example above w Rv R You would check 0 and 2 in the left set of check boxes a general 3d rotor has a non empty grade 0 and 2 1 in the right set a 3d vector has a non empty grade 1 and then click add This would optimize the product Rv The combination 2 4 ORDER 11 will appear in the list
67. ory for each multivector to store its coordinates But memory allocation costs computation time If memory has to be reallocated for each basic product or sum or other operation performance might suffer That is why Gaigen allows you to make a compromise between minimal memory usage and minimal memory allocation computation time You can choose from four memory allocation schemes ranging from lowest memory usage to lowest computational time e Tight Exactly the right amount of memory is allocated to store the co ordinates of the non zero grades of a multivector This implies frequent memory reallocation which is done via a simple and efficient memory heap e Balanced To prevent abundant memory reallocation the balanced allo cation scheme does not reallocate when less memory is required to store the coordinates up to a certain waste factor which the user can specify Suppose a variable holds a 3D rotor 4 coordinates and is assigned a vector 3 coordinates the memory waste would be 33 3 if 4 coordi nates memory locations would still be used to store 3 coordinates The balanced memory allocation algorithm then decides to either release the 4 old memory locations and allocate 3 new memory locations or to just waste 1 memory location This depends on the the waste factor If the number of used memory location divided by the number of required mem ory locations will become larger than or equal to the waste factor the al gorithm will de
68. place gaigenhl cpp who want to squeeze the last bit of performance out of their ap plication or those who have to change or maintain the internal C class Many functions of the internal C class return void unlike functions of the high level C interface which return the result Instead most internal C class functions store the result in the calling class e g c add a b adds a and b and stores the result in c 4 1 C Class The storage parts of the C class declaration looks something like this class e3gai Depending on the memory allocation algorithm Either a pointer or an array of floats float c pointer float c 8 array of floats bitfield for grade part and memory usage administration int usage information about the class static const int dim static const int nbCoor 43 44 CHAPTER 4 LAYER 1 INTERNAL C CLASS memory usage grade part usage mem usage 3 2 1110 msb Isb Figure 4 1 Storage of grade part and memory usage in the usage field the basis vectors and inverse pseudoscalar static e3gai el static e3gai e2 static e3gai e3 static e3gai bv 3 static e3gai I static e3gai Ii First of all the class contains either a pointer to an array of floating point vari ables or an array of floating point variables In either case it is called c for co ordintes The type used depends on the memory allocation algorithm selected in the storage tab
69. pplication retrieve the wrong coordinates get the scalar coordinate of a multivector float scalar const get the GRADEX coordinates of a multivector const float e3ga operator int grade const const float e3ga coordinates int grade const You can retrieve the largest coordinate of a multivector using largestCoor dinate 36 CHAPTER 3 LAYER 2 HIGH LEVEL C INTERFACE Get the fabs value of the absolutely largest coordinate of a multivector float e3ga largestCoordinate const For example e3ga a GRADE1 1 0 2 0 3 0 GAIM_ FLOAT lca a largestCoordinate lca 3 3 14 Coordinate string parsing Gaigen can parse its own string output and a bit more The function parseS tring takes as input a string describing a multivector as formatted for in stance by the string function parses it and sets the value of the multivector accordingly int e3ga parseString const char str const c3ga_ben ben NULL Usage example e3ga a int nb nb a parseString 1 0 e1 2 0 e1 e2 e3 The function returns the number of characters read from str or 1 on failure The optional second argument of parseString describes optional extra Ba sis Elements Names and the start and end delimited of the string These are contained in a small C class class e3ga_ben public e3ga_ben e3ga_ben char startDelimiter char endDelimiter e3ga_ben add a basis element name and
70. r const e3ga amp a const e3ga amp e3ga operator const e3ga amp a e3ga amp add const e3ga amp a const e3ga amp b subtraction 3 7 REVERSE CLIFFORD CONJUGATE AND GRADE INVOLUTION 29 e3ga amp e3ga operator const e3ga amp a const e3ga amp e3ga operator const e3ga amp a e3ga amp sub const e3ga amp a const e3ga amp b negation e3ga amp e3ga operator const e3ga amp e3ga negate const Note that the and operators which are used to decrement and incre ment integer and floating point variables by one in C have nothing to do with addition of subtraction for multivector variables They are used for the Clifford conjugate and the grade involution in Gaigen see section 3 7 3 7 Reverse Clifford Conjugate and Grade Involu tion The reverse Clifford conjugate and grade involution all toggle the sign of cer tain grade parts They each have a unary operator The and operators can be applied both pre and post fix e3ga a b a b a b a b gradeInvolition all of which are equivalent They do not alter the operand like the standard and operators for integer and floating point variables do The definitions of these functions and operators are veverse e3ga amp e3ga operator const e3ga amp e3ga reverse const clifford conjugate e3ga amp e3ga operator const e3ga amp e3ga operator int const e3ga amp e3ga c
71. r the maximum number of coordinates used by a multivector variable and the basis vectors and inverse pseuodscalar The basis vectors are available individually named as they in the signature tab and from an array of length d the dimension of the algebra The pseudoscalar ln is the dimension of the algebra plus 1 2One heap is available for each array length from 1 up to 2 4 2 CONSTRUCTORS ASSIGNMENT FUNCTIONS 45 and inverse pseudoscalar are always called I and Ii All these static multivector variables are initialized to their proper values when your application starts 4 2 Constructors Assignment Functions The constructors and assignment functions are identical to those used for the high level C interface except that the operator symbol is not available The random randomBlade and randomVersor are available 4 3 The Products The product functions like gp lcont and op all have the same syntax void e3gai gp const e3gal Sa const e3gai amp b void e3gai op const e3gai amp a const e3gai amp b void e3gai lcont const e3gai amp a const e3gai amp b void e3gai rcont const e3gai amp a const e3gai amp b void e3gai hip const e3gai amp a const e3gai amp b void e3gai mhip const e3gai amp a const e3gai amp b void e3gai scp const e3gai amp a const e3gai amp b They are used differently than those from the high level C interface the following line c gp a b compute
72. r integers in C The join isl 5 or operation for subspaces grade part operator selects the specified et a part of a multivector variable grade part The operator returns a pointer to an selection AERDEN array of floating point values repre fl Sen the coordinates for the speci a GRADE1 1 fied grade part of a multivector vari f2 able a GRADE1 2 We now discuss how to use the C high level interface Reading this sec tion and tutorial 1 which demonstrates the e3ga algebra which stands for Eu clidian 3 dimensional Geometric Algebra will give you a good understanding of how to write your own application using geometric algebra as implemented by Gaigen We will assume the classname is e3ga but in your application it might be any name since you can set it in the Gaigen user interface see sec tion 2 1 3 1 Construction To create a new multivector variable a with the value 0 you would use e3ga a If you want a scalar valued multivector variable b use 22 CHAPTER 3 LAYER 2 HIGH LEVEL C INTERFACE e3ga b 1 0 which creates a variable b with the value 1 0 To explicitly set the coordinates of a homogenous multivector variable a multivector for which all grade parts are O except for one or a blade you can use this constructor e3ga c GRADE1 1 0 2 0 3 0 e3ga t GRADE3 8 0 This creates a vector valued multivector c with the value le 2e2 3e3 and a multivector variable t with the value 8e
73. rade are always packed together in the coordinate array As an example if you change the order and orientation of the grade 2 basis blades of a 3d algebra to ez A ez ez A er e1 A e2 then the coordinates of the multivector A would be stored as 1 2 3 4 7 6 5 8 This order and orien tation is used by the pregenerated e3ga algebra and is the one shown in figure 2 2b 2 5 Functions The functions tab figure 2 3a simply contains a lot of checkboxes which can be used to include certain functions into the algebra code Some functions are interdependant on each other some only work for specific dimensions and most functions require that a product usually the geometric product outer product scalar product or left contraction is included in the algebra While you are still unexperienced with Gaigen and while Gaigen isn t finished yet it might be best to always include these 4 products in your algebra because 2 5 FUNCTIONS 13 fa Geometrac Algebra linglement ation Gener ater version 0 95 Algebra Implement ation Generator version 0 95 Qeneral signature products order functions storage generate general signature products order functions storage generate o A mere ocecanate Storage loading point type Sigt Noat grade involuton nom project and reject bared double e reverse nome random o mapo caora conjugate versor rerse tast temporary variables 0 max negate loum
74. represented by its own class The of this class is the name of your algebra class e g e3ga with om concatenated to it e g e3gaiom You can initialize the outermorphism operator when you construct it or later on using one of the init functions The outermorphism is initialized by either passing it the images of all basis vectors under the linear transformation or by passing it a spinor If you pass a spinor the initialization function will assume that the right way to apply the spinor is e3ga spinor vector vectorImage vectorImage spinor vector spinor 32 CHAPTER 3 LAYER 2 HIGH LEVEL C INTERFACE It will compute the vector images of all basis vectors and then pass these to the vector images initilization function A typical use of the outermorphism operator would be something like this const int nb 1000 int i e3ga rotor lotsOfVectors nb lotsOfVectorImages nb e3ga_om om initialize the outermorphism operator om initSpinor rotor apply it to the vectors for i 0 i lt nb i lotsOfVectorImages i om lotsOfVectors i this is the other way to compute the vector images lotsOfVectorImages i rotor lotsOfVectors i rotor GRADE1 57 Of course the outermorphism operators doesn t work on vectors only you can also use it to transform blades of any grade The outermorphism operator internally constructs a matrix representation for the linear operator The
75. roduct The abbreviations are also used inside the generated source code as names of the functions which compute the products The products tab is used to select which of these products you want to include in your algebra A common selection would be geometric outer and scalar product and either the left contraction or one of the Hestenes inner prod ucts To include a product in your algebra click on the tab of that product and check the checkbox left of the full name of the product The color of the text in the tab will change from black to red to reflect the inclusion of that product Below the checkbox is a large field that is used to control the optimizations implemented for each product You are not required to add any optimizations to generate a basic algebra but you can significantly increase the performance of your application by doing so sometimes by an order of magnitude But by making the wrong optimizations you could in theory decrease the perfor mance of your application So it is important that you understand how the optimizations work if you care about performance The optimizations are based on the assumption that it is likely that your application will use certain products between certain multivectors much more often than others Suppose you use lots of rotations of 3D vectors in your program like w Rv R 2 1 Then the performance of your program could be increased if Gaigen would generate an optimized function f
76. s the geometric product of a and b and stores it in c No temporary variables are used Versions of the outer product function are available in which one argument can be a floating point value void e3gai op const e3gai amp a float scalar void e3gai op float scalar const e3gai amp a All product functions first check to see if an optimized function exists to compute the product of the arguments If so they set the grade part and mem ory usage of the result and call the optimized function Otherwise they expand the arguments to arrays of pointers to arrays of floating point variables like the one shown in figure 5 1 call the general implementation of the function and compress the result into the multivector variable 4 3 1 Euclidean Metric Products Only when the meet and join or factoring functions are included in algebras with a non euclidean signature special euclidean implementations of some metric products geometric product left contraction and scalar product are available They are used to implemented a LIFT see 5 Their definitions are void e3gai gp em const e3gai amp a const e3gai amp b void e3gai lcont_em const e3gai Sa const e3gai amp b void e3gai scp em const e3gai ta const e3gai amp b 46 CHAPTER 4 LAYER 1 INTERNAL C CLASS 4 4 Other Functions To add or subtract two multivectors use void e3gai add const e3gai amp a const e3gai amp b void e3gai sub const e3gai amp a cons
77. s which square to O with itself but to 1 or 1 with the other A pair of null vectors with these properties can be created by checking the reciprocal checkbox between two basis vectors An example of the use of reciprocal null vectors are ey and e in the conformal model Gaigen can support these directly The dropdown box to the right of the checkbox can then be used to select the value the pair of vectors squares to Because a reciprocal checkbox is only provided between neighboring basis vectors so only a limited set of reciprocal null vectors can be created like this This is not just a user interface limitation but a limitation in the way the code that Gaigen generates works It is not a limitation of geometric algebra in general but we believe an algebra specification can always be modified slightly to archive the same result with neightboring reciprocal null vectors 2 3 Products Currently Gaigen supports seven basic products between multivectors Be sides these basic products some special products are available such as the outermorphism meet and join and the delta product But only the seven basic products are controlled from the products tab which is shown in figure 2 2a The products tab itself contains another set of tabs one for each product The names of these tabs are abbreviated versions of the products 10 CHAPTER 2 THE USER INTERFACE ep Geometric Product Right Contraction op Outer Product Scalar P
78. se we wanted performance similar to optimized hand written code while maintaning full generality for scientific research and experimentation many geometric algebras with differ ent dimensionality signatures and other properties may be required Instead of coding each algebra by hand Gaigen provides the possibility to generate the code for exactly the geometric algebra the user requires This code may be less efficient than fully optimized hand written code but is likely to be much more efficient than one library that tries to support all possible algebras at once One such a library CLU 6 uses C templates and style classes which specify the properties of the geometric algebra While the CLU approach has its advantages e g special code can easily be added to the style classes that represent an algebra and the size of the code implementation may be smaller especially when more than 1 algebra is used initial experiments show that the performance of code generated by Gaigen is about an order of magnitude more efficient in terms of computation time Another disadvantage of CLU is that for each algebra you have to write a new style file four style files are provided in the package you have to write your own style file In Gaigen you simply specify what you want in the user interface hit the generate button and you can use your new algebra The Gaigen package consists of e this manual which describes how to use Gaigen and what its
79. t e3gai amp b To add a multivector and a floating point variable these functions are avail able void e3gai add void e3gai add void e3gai sub void e3gai sub float scalar const e3gai b const e3gai amp b float scalar float scalar const e3gai b const e3gai amp b float scalar ZA These functions compute the reverse clifford conjugate and grade involu tion of a multivector and store it in the calling multivector void e3gai reverse const e3gai ta void e3gai cliffordConjugate const e3gai ta void e3gai gradeInvolution const e3gal amp a The three flavours of the inverse can be used with these functions compute inverse for versors only int e3gai versorInverse const e3gal ta compute inverse for general 3d multivectors int e3gai lounestoInverse const e3gai ga compute inverse for general multivectors uses gaussian elimination slower and less stable than versorInverse int e3gai generalInverse const e3gai ta These four functions can be used to extract grade parts or to get informa tion about the grade of a multivector void e3gai takeGrade const e3gai amp a int grade int e3gai highestGrade const e3gai amp a int e3gai grade const int e3gai maxGrade const takeGrade extracts a single grade part from a multivector highestGrade ex tracts the highest non zero grade part from a multivector and returns its grade index grade returns the gra
80. tal compiler uses LAPack to compute the general products The opt2c2 compiler splits up the products into many little functions and lets the product functions call those as required 54 CHAPTER 5 LAYER 0 LOW LEVEL COMPUTATIONAL FUNCTIONS Chapter 6 File Formats 6 1 gaspectemplates txt file Gaigen reads and copies most piece of code that it generates from the gaspectem plates txt file The syntax of that file is discussed in this section To read parse and output the code contained in the file the codeTemplateContainer class in codetemplate cpp and codetemplate h is used The gaspectemplates txt file consists of codeblocks Codeblocks are blocks of code which Gaigen copies into the source code it generates In early versions of Gaigen all code was simply printed to the sourcefiles using fprintf This be came very messy since all of the code that now resides gaspectemplates txt over a thousand lines was contained in gaspec cpp Instead of that we put almost all of the code in templates or codeblocks in gaspectemplates txt That makes them easy to read create and maintain Only the code which is very difficult to copy and paste e g which is extremely dependent on the dimension of the algebra is printed directly from gaspec cpp Of course Gaigen does not literally copy code from gaspectemplates txt Many parts of the generated code depend on the dimension name floating point type and many other properties of the require
81. ultitheading Multitheading support has not yet been fully implemented in Gaigen The main functions which are not mt safe are part of the tight and balanced mem ory allocation algorithms the fast temporary variable allocation and the string function Use of the string and print and fprint functions can be avoided if you really want to use multithreading and the max or maxpp memory alloca tion algorithms can be used Multitheading support might be more complete in the future 3 21 Drawing There are no multivector drawing functions available in Gaigen We have im plemented GAViewer that allows you to view multivectors interactively with minimal intrusion in your application Your application only has to be able to write multivectors and commands to a file which can be conviniently done using the print functions of Gaigen This allows you to view results of com putions and to debug your application without adding an OpenGL UI to it Chapter 4 Layer 1 Internal C class This chapters describes the internal C class The source code for this class is generated by Gaigen Much of this generated source code is copied and pasted with adjustments from gaspectemplates txt a file which is described in section 6 1 Most people will not want to use the internal C class to actually pro gram applications since it s a rather crude interface This chapter is intended for people who want to create their own high level C interface to re
82. vailable from the high level C interface Using the basis vectors and inverse pseudoscalar you can write things like e3ga a b one a e3ga el e3ga e2 b a lt lt e3ga Ii one e3ga li e3ga 1 The definitions of the basis vectors and inverse pseudoscalar are static e3gai e3gai el static e3gai e3gai e2 static e3gai e3gai e3 static e3gai e3gai I pseudoscalar static e3gai e3gai li inverse pseudoscalar static e3gai e3gai bv 3 array of pointers to el e2 bv is an array of pointers to the basis vectors The basis vectors e1 e2 and e3 in the example of the e3ga algebra can have any name In the 5d c3ga algebra for instance the basis vectors are called el e2 e3 e0 einf a 3d color algebra could name the basis vectors red green and blue To change the name of the basis vectors use the signature tab section 2 2 Static in this context means that there is only one global copy of such a variable it is not included in every instantiation of a multivector variable e3 40 CHAPTER 3 LAYER 2 HIGH LEVEL C INTERFACE 3 18 Use of Internal Class and Floating Point Vari ables in Functions Besides writing statements like e3ga a b C c a b you of course also want to write e3ga a b C c 2 0 a b and e3ga a b C c e3ga el a e3ga e2 even though 2 0 e3ga el and e3ga e2 are not variables of type e3ga 2 0 is of type float or double e3g
83. wback Since the array of temporary variables has a fixed size at some point old temporary variables will have to be recycled If these old temporary variables are still in use somewhere in you program their value will be overwritten This will lead to bugs in you program which are very hard to find especially of you don t know what the symptoms of the problem are The section on temporary variables in tutorial 1 3 demonstrates the problem The rule of thumb is to never keep a reference to a temporary multivector variable when you have enabled fast temporary variables in the functions tab The two main ways to keep a reference to a temporary variable are passing them on to another function like this someFunction a b and explictly keeping a reference e3ga amp C a b Explictly keeping a reference can be easily avoided and pass a temporary vari able to a function can be prevented like this Cc a b someFunction c If you have a look at gaigenhl h and gaigenhl cpp you will see that functions and operators return the type GAIM_RETURN TYPE This is actually a macro defined as either e3ga or e3ga amp depending on whether the fast temporary variables are enabled or not The allocation of temporary variables is managed by the GAIM_RETURN_VAR variableName macro Both these macros are defined in gaigenhl h this size is defined by MV_MAX_TEMP in gaigenhl h 42 CHAPTER 3 LAYER 2 HIGH LEVEL C INTERFACE 3 20 M

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