The Power of Geometric Algebra Computing: For Engineering and Quantum Computing 1st Edition Dietmar Hildenbrand
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The Power of Geometric Algebra Computing The Power of Geometric Algebra Computing Dietmar Hildenbrand
MATLAB • is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB • software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB • software.
First edition published 2022 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
© 2022 Dietmar Hildenbrand
CRC Press is an imprint of Taylor & Francis Group, LLC
Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint.
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Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe.
ISBN: 978-0-367-68458-7 (hbk)
ISBN: 978-0-367-68775-5 (pbk)
ISBN: 978-1-003-13900-3 (ebk)
DOI: 10.1201/9781003139003
Typeset in Nimbus Roman by KnowledgeWorks Global Ltd.
Figure1Zeelandimpression.
CHAPTER 17 GAALOPWebasaQubitCalculator 163
17.1QUBITALGEBRAQBA163
17.2GAALOPWEBFORQUBITS164
17.3THENOT-OPERATIONONAQUBIT165
17.4THE2-QUBITALGEBRAQBA2165
CHAPTER 18 Appendix169
18.1APPENDIXA:PYTHONCODEFORTHEGENERATIONOF OPTIMIZEDMATHEMATICACODEFROMGAALOP169
ListofFigures 1Zeelandimpression.
1.1Screenshotofhttps://bivector.net. 2
2.1Theintersectionoftwospheresresultsinacircle. 10
2.2Translationofaspherefromtheorigintothepoint Pt 12
2.3Vis2dVisualizationoftheperpendicularbisectorbetweenthetwo (red)points. 15
2.4Thebasicsofthe2DProjectiveGeometricAlgebraaccordingto[13].18
2.5Geometrybasicsofthe2DProjectiveGeometricAlgebraaccording to[13]. 19
2.6VisualizationoftheListing2.1. 20
2.7Geometrybasicsofthe3DProjectiveGeometricAlgebraaccording to[13]. 21
2.8Normsandmotorsofthe3DProjectiveGeometricAlgebraaccordingto[13]. 22
2.9VisualizationoftheListing2.2. 23
3.1GlobalSettingPluginfortheconfigurationofMaxima(aswellas fontsizes). 26
3.2ConfigurationofGAALOPforvisualizationsbasedonCompass RulerAlgebra. 26
3.3Screenshotoftheeditor,thevisualizationandtheoutputwindowof GAALOP. 27
3.4SpheresandlinesarebasicentitiesofGeometricAlgebrathatone cancomputewith.Intersectionoftheseobjectsareeasilyexpressed withthehelpoftheirouterproduct.Here,oneofthetwopointsof theintersectionofthesphereandthelineisshowninblue.32
4.1TheprogramminglanguageselectionofGAALOPWeb. 33
4.2ThescreenofGAALOPWebforPython. 34
4.3 36 xiii GAALOPWebvisualizationofthebisectorexample. v
4.4GAALOPWebvisualizationoftherotationofacircle. 37
4.5VisualizationoftheListing4.4withfourpointsdefiningasphere.39
5.1Thesphere-centreexamplewithGAALOPWeb. 42
5.2Theresultofthesphere-centerexample. 42
5.3Thesphere-centreexamplewithouttheP1zvariable. 43
5.4Theresultofthesphere-centerexamplewithadefaultsliderforthe P1zvariable. 43
5.6Theresultofthesphere-centreexamplewitha fortheP1zvariable. 44
6.1Theline-sphereexamplewithGAALOPWeb. 50
6.2Theresultoftheline-sphereexamplewithGAALOPWeb. 51
6.3Jupyternotebookoftheline-sphereexamplebasedonthePython ConnectorofGAALOPWeb(PartI). 52
6.4Jupyternotebookoftheline-sphereexamplebasedonthePython ConnectorofGAALOPWeb(PartII). 52
6.5Jupyternotebookforthevisualizationoftheline-sphereexample.53
6.6Simplevisualizationofsomegeometricobjectswithpyganja.54
6.7Pyganjascenegraphvisualization. 55
6.8Jupyternotebookoftheline-sphereexamplebasedonthePython ConnectorofGAALOPWeb(PartII). 56
7.1FiguregeneratedinMathematicafortheexampleabove.Thepoints inblackareexactlythepointsfoundbytheCGAfunction.64
7.2ScreenshotofGAALOPWeb. 65
7.3OutputofGAALOPWeb. 66
8.1Manipulatormodeldescription. 73 8.2 80
ThescreenofGAALOPWebforMATLAB.
Motionoftherobot:left–MATLAB-optimizedcode,right–ABB user-definedslider Thesphere-centreexamplewithauser-definedsliderdefinition.
8.3 RobotStudio.Thisshowsthatthefunctionalityofbothalgorithms isidentical. 81
8.4Measuringoftheruntimethroughthewholerobot’smotion.82
10.1Anellipsetobevisualized. 90
10.2ThevisualizationofanellipsewithGAALOPWeb. 91
10.3TheresultofListing10.5. 92
10.4GAALOPWebvisualizationoftwointersectinglinesbasedon5 points.93
10.5GAALOPWebvisualizationofanellipsebasedon5points.94
10.6GAALOPWebvisualizationofahyperbolabasedon5points.94
10.7GAALOPWebvisualizationofsometransformationsofanellipse.96
10.8GAALOPWebvisualizationfordilationsofconics. 97
10.9Visualizationoftheintersectionofellipseandcircle.
10.10VisualizationresultofListing10.12.
11.1Vis3Dsettings.
11.2Vis3Dvisualizationofellipsoid,toroidandsphere.
11.3Vis3Dvisualizationofplaneandline.
11.4Vis3Dvisualizationofcylinders.
11.5Vis3Dvisualizationofcones.
11.6Vis3Dvisualizationofparaboloids.
11.7Vis3Dvisualizationofanhyperboloid.
11.8Vis3Dvisualizationofcyclides.
11.9Vis3Dvisualizationoftransformationsofanellipsoid. 120
11.10Vis3Dvisualizationoftheintersectionofanellipsoidandaline.121
11.11Vis3Dvisualizationofconicsinspace. 122
11.12Vis3Dvisualizationofthereflectionandprojectionofalineontoa sphere. 123
11.13Vis3Dvisualizationoftheinversionofatoroid. 124
12.1Vis3Dvisualizationofasimplecubiccurve.
129
13.1ReflectionofasphereaccordingtothealgorithmofListing13.1.132
13.2Screenshotof“GAALOPWebforGAPP”. 133
14.1GAPPCOco-processorwithhostinterface.
14.2GAPPunitprocessinganinputVector(vectorwithallthescalar inputvalues)toresultingvectors.
136
137
14.3GAPPunitconsistingof2levelsofDotVectorsunits. 137
14.4Paralleldotproductoftwo n-dimensionalvectorsVector0andVector1(n parallelproductsfollowedbylog(n)paralleladditionsteps).138
14.5ConfigurableGAPPCOIblockdiagram.
14.6DotVectors1blockdiagram.
138
139
14.7GAPPunitconfiguredtoexecutereflectionoperations. 140
14.8InternalconfigurationdatastructureforGAPPCObasedonthereflectionexample. 141
14.9Screenshotof“GAALOPWebforGAPPCO”. 144
15.1ThegeneralGAPPCOIIdesign.
15.2Theparallelcomputinglayer.
15.3MappingofaDotVectorsunitintotheGAPPCOIIdesign.148
15.4Step1.
15.5Step2.
15.6Step2A.
15.7Step1.
15.8Step2.
15.9Step3.
15.10Differentprocessingtimes.
16.1Statevector ψ ofaquantumregisterbuiltfromtwoquantumbits accordingto[41]. 156
16.2Reflectionaboutthereflectionvector. 158 16.3GraphicalillustrationoftheHadamardtransformaccordingto[41].160
17.1ScreenshotofGAALOPWebforQubitalgebras.
ListofTables 2.1NotationsofGeometricAlgebra
2.3The32basisbladesofthe5DConformalGeometricAlgebra(ConformalGeometricAlgebra)
2.4Thetworepresentations(IPNSandOPNS)ofconformalgeometricentities.Theyaredualsofeachother,whichisindicatedbythe asterisksymbol.
2.5GeometricMeaningoftheInnerProductofPlanes,Spheresand Points
2.6The16BasisBladesoftheCompassRulerAlgebra(tobeidentified bytheirindices)
2.7TheRepresentationsoftheGeometricObjectsoftheCompass RulerAlgebra 15
2.8GeometricMeaningoftheInnerProductofLines,CirclesandPoints16
2.9TheDescriptionofTransformationsofaGeometricObject o in CompassRulerAlgebra(pleasenotethat LoL meansthegeometricproductof L, o and L) 16
3.3MacrosofGAALOPScriptforConformalGeometricAlgebra30 3.4PragmasofGAALOPScript
7.1TheResultsfortheFunction DGPSolver [61]RunningtheConventionalandtheCGAapproach.
8.1ComputationTimeofAlgorithmsin(s)
14.1ConfigurationbitsforDotVectors1ofFig.14.6.(Aregisterfilecomposedof3232-bitregistershasbeenconsideredforthisfirstdesign andtherefore5-bitaddressesareneeded.)
14.2ConfigurationBitstreamofGAPPCOI143
14.3ConfigurationBitstreamforGAPPunitRealizingreflectorFunctionality145
17.1The4BasisVectorsoftheQubitAlgebra
17.2The8BasisVectorsofthe2-QubitAlgebraQBA2
Foreword Geometricalgebrashaveprovedtobeusefulinengineeringapplicationsduringthe pastthreedecades.ItallbeganwithConformalGeometricAlgebra(CGA)asamodel ofthree-dimensionalEuclideanspaceanditstwo-dimensionalversion,Compass RulerAlgebra(CRA).Boththeseoutstandingalgebrasfromthecomputingpoint ofviewwereatopicfortheprevioustwobooksbyDietmarHildenbrand.Itisonly naturalthathisnewbookistreatingnewlyreinventedstructures,suchasProjective GeometricAlgebra(PGA),GeometricAlgebraforConics(GAC)oraCliffordalgebraforquantumcomputing,QuantumBitAlgebra(QBA).Theunitingelementin allbooksbyDietmarHildenbrandisGAALOPoritsonlineversionGAALOPWeb. Indeed,astherangeofapplicationsisgrowing,thereisaneedforatoolthatissimplyimplementingsymbolicoperationsinparticularCliffordalgebrasintoacodethat canbeeasilyincludedinstandardprogramminglanguagesorengineeringsoftware tools.Furthermore,ifageometricalgorithmisprecompiledbyGAALOP,alloperationsareoptimizedwithrespecttothecomputationalcomplexity.Theadvantage ofsuchanapproachliesalsointhefactthatthedimensionofaCliffordalgebrais notsuchaseriouslimitationregardingthenumberofoperationsandcomputational load.Andfinally,thegeometricnatureofthealgorithmsdemandsgeometricimaginationwhichhastobeverified.Thus,GAALOPWebprovidesavisualiszationtool basedonaPythonmodule,Ganja,wherethegeometricideasmaybecheckedinstantly.Analogouslytothepreviousbooks,eventhisoneiseasytoreadbecausethe authorprovidesexamplesofapplicationsforparticulartopics.Togetherwithsymboliccalculations,thereaderfindscodeinthelanguageforGAALOPWeb,which isGAALOPScript.Therefore,allapplicationsmaybesimplychecked,geometric primitivesandoperationscalculatedandvisualized.Specialfocusisonquantum computingwhereGAALOPWebservesalsoasaq-bitcalculator.Thisisanovel conceptwhichdeservesattentionandGAALOPWebagainproveditsuniversalityin accommodatingdifferentstructures.AfterintroducingthebasicsofgeometricalgebracomputingforCGA,CRAandPGA,thebookcontinueswiththedescription ofGAALOPWebfunctionalities,includingvisualization.Consequently,fourpossibilitiesofgeneratingacodeforparticularprogramminglanguagesandengineering toolsaredescribedintheirrespectivechapters.ThisincludesC/C++,Python,MathematicaandMATLAB R ,respectively.Eachdescriptionisaccompaniedbyspecific engineeringapplicationscontainingthecodeinGAALOPScriptforinstantverificationinGAALOPWeb. Chapters8 to 10 summarisethepowerofGAALOPWebwhen usedforhigh-dimensionalgeometricalgebras,suchasDoubleConformalGeometricAlgebra,GeometricAlgebraforConicsandforCubics.Subsequenttwochapters
dealwithGAPP,theintermediatelanguageofGAALOPanditsimplementationin GAALOPWeb.Thisisverysuitableforhardwareimplementations;oneofthemis theGeometricAlgebrahardwaredesignGAPPCOIand,newly,GAPPCOII. Chapter15 andtheremainingchaptersintroducetheconceptofquantumcomputingin ageometricalgebrasetting.Eveninthisarea,GAALOPWebproveditsqualityand isusedasaq-bitcalculator.Asasummary,Ibelievethatthisbookmaywellserve tosupportgeometricalgebrasinpenetratingengineeringapplications,yetitissuitableeveninacademiawhereitcanguidebothstudentsandteachersinafancyway. Indeed,thepotentialofgeometricalgebrasmaybeexploitedifandonlyifitcanbe introducedtoawideaudience,andtodoso,thecalculationsmustbesimpletoimplement,possibletobeincludedinstandardengineeringsoftwaretoolsandvisualized. Andthisispreciselywhatthisbookisabout.Itismypleasurethatthebookwillbe introducedontheoccasionofthe8thConferenceonAppliedGeometricAlgebrasin ComputerScienceandEngineering(AGACSE)inBrno,CzechRepublic,2021.
Prof.PetrVasik
BrnoUniversityofTechnology
Brno,CzechRepublic April1,2021
Preface GeometricAlgebraisaverypowerfulmathematicalsystemforaneasyandintuitive treatmentofgeometry,butthecommunityworkingwithitisstillverysmall.The maingoalofthisbookistoclosethisgapfromacomputingperspectiveinpresenting thepowerofGeometricAlgebraComputingforengineeringapplicationsandquantumcomputing.Theintendedaudienceincludesstudents,engineersandresearchers interestedinreallycomputingwithGeometricAlgebra.
Today,weindeedhavetheGeometricAlgebraComputingtechnologyavailable foreasy-to-develop,geometricallyintuitive,robustandfastengineeringapplications. Wearehappytoprovide GAALOPWeb,theweb-basedversionofour GAALOP (GEOMETRIC ALGEBRA ALGORITHMS OPTIMIZER) compilerfortheintegrationofGeometricAlgebraintostandardprogramminglanguages.BesidesthegenerationofoptimizedcodeforlanguagessuchasC++,Python,Mathematica,MATLAB andmanyothers,GAALOPWebcanalsobeusedfortheconfigurationofaspecific GeometricAlgebrahardware,theGAPPCO1 co-processor.ItisalsoabletovisualizeGeometricAlgebraalgorithms.GAALOPWebcanbeusedeasilyonPC,smart phone,tabletetc.withoutanysoftwareinstallation.
While ThePowerofGeometricAlgebraComputing reallyfocusesontheeasy-tohandlecomputingwithGeometricAlgebra,thereisafollowingrelationtomyformer books: FoundationofGeometricAlgebraComputing[29]focusesontheGAALOP technologyitselfandon3Dapplicationsforcomputergraphics,computervisionand robotics.Thesecondbook IntroductiontoGeometricAlgebraComputing[30]isintendedtogivearapidintroductiontocomputingwithGeometricAlgebra.Fromthe pointofviewofgeometricobjects,itfocusesonthemostbasicones,namelypoints, linesandcircles.Wecallthisalgebra CompassRulerAlgebra,sinceyouareableto handleitcomparablytoworkingwithacompassandruler.Itoffersthepossibility tocomputewiththesegeometricobjects,theirgeometricoperationsandtransformationsinaveryintuitiveway.Whilefocusingon2D,itiseasilyexpandableto3D computationsasusedinmanybooksdealingwiththepopularConformalGeometric Algebra.
ThepowerofGeometricAlgebraComputingcanalsobestudiedinthisbookby usinghighdimensionalalgebrasforthehandlingofmorecomplexgeometricobjects suchasconics,quadricsandcyclides.Last,butnotleast,theadvantageoususeof GeometricAlgebraforquantumcomputingisdemonstrated.
1Geometricalgorithmsparallelizationprogramsco-processor
IreallydohopethatthisbookcansupportthewidespreaduseofGeometricAlgebraasamathematicaltoolforcomputingwithgeometryinengineeringapplications aswellasinquantumcomputing.
Dr.DietmarHildenbrand
Acknowledegments IwouldliketothankDr.-Ing.ChristianSteinmetzforhistremendoussupporttothis book.HedevelopedGAALOPWebasusedinthisbookduringhisPhDphase. Manythanksto
-thegroupofProf.PetrVasikfromBrnoUniversityofTechnologyfortheir supportconcerningtherobotapplication,theGeometricAlgebraofconicsand quantumcomputing,
-Prof.RafaelAlves(UniversidadeFederaldoABC,Brasil)forhissupport regardingthemoleculardistanceapplicationaswellastheapplicationofGeometricAlgebratoquantumcomputing,
-Dr.SilviaFranchini,Prof.A.Gentile,Prof.G.VassalloandProf.S.Vitabile fromtheUniversityofPalermoforthecooperationregardingthenewcoprocessordesignGAPPCOI,
-RobertEasterforthefirstintegrationofDoubleConformalGeometricAlgebraintoGAALOPandhissupportwithtestscripts,
-SeniorAssociateProf.EckhardHitzerformanyfruitfuldiscussionsandfor hisbigeffortandenthusiasmforthepromotionofGeometricAlgebra(and computingwithit).
GeometricAlgebraisaverypowerfulmathematicallanguagecombininggeometric intuitivitywiththepotentialofhighruntime-performanceoftheimplementations. Thisbookonhandisbasedon GAALOPWeb,anewuser-friendly,web-basedtoolfor thegenerationofoptimizedcodefordifferentprogramminglanguagesaswellasfor thevisualizationofGeometricAlgebraalgorithmsforawiderangeofengineering applications.Itincludesapplicationsfromthefieldsofcomputergraphics,robotics andquantumcomputing.
Thebook FoundationsofGeometricAlgebraComputing [29]describes GAALOP(see Chapt.4)inaveryfundamentalway,sinceitbreaksdownthecomputingofGeometricAlgebraalgorithmsintothemostbasicarithmeticoperations.This bookonhandmakesuseofitsweb-basedextention GAALOPWeb (see Chapt.4).
ThisbookissuitableasastartingpointforcomputingwithGeometricAlgebra foreverybodyinterestedinitasanewpowerfulmathematicalsystem,especially forstudents,engineersandresearchersinengineering,computerscience,quantum computingandmathematics.
1.1GEOMETRICALGEBRA ThemainadvantageofGeometricAlgebraisitseasyandintuitivetreatmentofgeometry.
GeometricAlgebraisbasedontheworkoftheGermanhighschoolteacherHermannGrassmannandhisvisionofageneralmathematicallanguageforgeometry. Hisveryfundamentalwork,called Ausdehnungslehre[24],waslittlenotedinhis time.Today,however,Grassmannismoreandmorerespectedasoneofthemost importantmathematiciansofthe19thcentury.
WilliamClifford[11]combinedGrassmann’sexterioralgebraandHamilton’s quaternionsinwhatwecall Cliffordalgebra or GeometricAlgebra1.Pioneeringwork
1DavidHesteneswritesinhisarticle[27]aboutthegenesisofGeometricAlgebra: EventodaymathematicianstypicallytypecastCliffordAlgebraasthe“algebraofaquadraticform,”withnoawareness
hasbeendonebyDavidHestenes.Especiallyinterestingforthisbookishisworkon ConformalGeometricAlgebra(CGA)[26][50].
1.2GEOMETRICALGEBRACOMPUTING EspeciallysincetheintroductionofConformalGeometricAlgebra(see Sect.2.2) therehasbeenanincreasinginterestinusingGeometricAlgebrainengineering.The useofGeometricAlgebrainengineeringapplicationsreliesheavilyontheavailabilityofanappropriatecomputingtechnology.Themainproblemof Geometric AlgebraComputing istheexponentialgrowthofdataandcomputationscomparedto linearalgebra,sincethe multivector2 ofan n-dimensionalGeometricAlgebrais2ndimensional.Forthe5-dimensionalConformalGeometricAlgebra,themultivector isalready32-dimensional.
AnapproachtoovercometheruntimelimitationsofGeometricAlgebrahasbeen achievedthroughoptimizedsoftwaresolutions.ToolshavebeendevelopedforhighperformanceimplementationsofGeometricAlgebraalgorithmssuchastheC++ softwarelibrarygeneratorGaigen2fromDanielFontijneandLeoDorstoftheUniversityofAmsterdam[21],GMacfromAhmadHosneyAwadEidofSuezCanal University[20],theVersorlibrary[12]fromPabloColapintoandtheC++expressiontemplatelibraryGaalet[60]fromFlorianSeyboldoftheUniversityofStuttgart. BiVector.net(see Figure1.1)givesagoodoverviewoversoftwaresolutionsfor
GeometricAlgebra.ItprovidescodegeneratorsforC++,C#,Python,Rustaswellas startingpointsforlibrariesforPython,C/C++,JuliaandtheJavaScriptvisualization toolGanja(see Sect.2.4).
OurGAALOPcompiler[32]isnotspecificforoneprogramminglanguagebut supportsmanyofthem.ItcanbeusedasacompilerforlanguagessuchasC/C++,
ofitsgranderroleinunifyinggeometryandalgebraasenvisagedbyCliffordhimselfwhenhenamedit GeometricAlgebra.IthasbeenmyprivilegetopickupwhereCliffordleftoff—toserve,sotospeak,as principalarchitectofGeometricAlgebraandCalculusasacomprehensivemathematicallanguagefor physics,engineeringandcomputerscience.
2ThemainalgebraicelementofGeometricAlgebra(pleasereferto Sect.2.2)
C++ AMP, OpenCL and CUDA [29] [31] as well as Python,
MATLAB, Mathematica,
Julia or Rust. Please find details about GAALOP in Chapt. 3 and about its web-based extensionGAALOPWebin Chapt.4.
1.3OUTLINE Chapt.2 presentsthemostimportantGeometricAlgebrasforengineering,namely ConformalGeometricAlgebra(CGA),CompassRulerAlgebra(CRA)andProjectiveGeometricAlgebra(PGA).
Chapt.3 isdealingwith GAALOP and Chapt.4 with GAALOPWeb,ournew user-friendly,web-basedversionofGAALOPforawiderangeofengineeringapplicationsbasedonGeometricAlgebraalgorithms.Itmakesitmucheasierforusers togenerateoptimizedsourcecodewithoutanysoftwareinstallation.GAALOPScript,thelanguagetodescribeGeometricAlgebraalgorithmsforthehandlingwith GAALOPWebispresentedin Sect.3.2.GAALOPWebsupportstheuserwithvisualizationsofGeometricAlgebraalgorithmsasdemonstratedin Sect.4.3.ThevisualizationisbasedonGanjaasdescribedin Sect.2.4.
InthefollowingchapterssomeapplicationsofGAALOPWebfordifferentprogramminglanguagesarepresented. GAALOPWebforC/C++ ispresentedin Chapt. 5 and GAALOPWebforPython in Chapt.6.AMolecularDistanceApplicationusing GAALOPWebforMathematica isshownin Chapt.7 andanapplicationofrobot kinematicsbasedon GAALOPWebforMatlab in Chapt.8.
Thepowerofhigh-dimensionalGeometricAlgebras,aspresentedin Chapt.9, comeswiththeirabilitytoeasilyhandlecomplexgeometricobjects.WhilethedefaultGeometricAlgebraofGAALOPWebisthe5DConformalGeometricAlgebra withpoints,spheres,planes,circlesandlinesasgeometricobjects,threeadditional algebrasarepresented.ThehandlingofconicsinGAALOPWebisshownin Chapt. 10.Quadricsasanotherexampleofgeometricobjects,whichareinterestingforapplications,arehandledin Chapt.11 andcubicsin Chapt.12
GeometricAlgebrahasaninherentpotentialforparallelization.Thiscanbevery wellseeninanintermediatelanguageofGAALOPcalledGAPP. Chapt.13 describesGAALOPWebforthislanguagewhichisverywellsuitableforhardware implementations. Chapt.14 shows,how GAALOPWebforGAPPCO isdealingwith theGeometricAlgebrahardwaredesignGAPPCO. Chapt.15 describesanewhardwaredesigncalledGAPPCOII.
AlsoquantumcomputingbenefitsalotfromGeometricAlgebra.Thisisduetoits abilitytodescribethequantumbit(qubit)operationsasgeometrictransformations. Thebasicsofquantumcomputingarepresentedin Chapt.16.Theprincipleofanew GeometricAlgebraforqubitstogetherwithsomeexamplesispresentedin Chapt. 17.WeextendedGAALOPWebforitinawaytouseitasaqubitcalculator.
GeometricAlgebrasfor Engineering ThischapterpresentsthebasicsofGeometricAlgebraaswellasthemostimportant GeometricAlgebrasforengineering,namelyConformalGeometricAlgebra(CGA), CompassRulerAlgebra(CRA)andProjectiveGeometricAlgebra(PGA).
2.1THEBASICSOFGEOMETRICALGEBRA ThemainproductofGeometricAlgebraiscalledthe geometricproduct;manyother productscanbederivedfromit.ThethreemostoftenusedproductsofGeometric Algebraarethe outer,the inner andthe geometric product.Thenotationsofthese productsarelistedin Table2.1.Wewillusetheouterproductmainlyfortheconstructionandintersectionofgeometricobjects,whiletheinnerproductwillbeused forthecomputationofanglesanddistances.Thegeometricproductwillbeused mainlyforthedescriptionoftransformations.
BasisBlades aretheelementaryalgebraicelementsofGeometricAlgebra.An n-dimensionalGeometricAlgebraconsistsofbasisbladeswith grades 0ton,where
