Geometric algebra computing: in engineering and computer science

549 Pages·2010·18.42 MB·English

by  David Hestenes (auth.)

Geometric Algebra Computing

Eduardo Bayro-Corrochano (cid:2) Gerik Scheuermann
Editors
Geometric
Algebra
Computing
in Engineering
and Computer Science
Editors
Prof.EduardoBayro-Corrochano Prof.Dr.GerikScheuermann
Dept.ElectricalEng.& Inst.Informatik
ComputerScience UniversitätLeipzig
CINVESTAV 04009Leipzig
UnidadGuadalajara Germany
Av.Científica1145 [email protected]
45015ColoniaelBajío,
Zapopan,JAL
Mexico
[email protected]
http://www.gdl.cinvestav.mx/edb
ISBN978-1-84996-107-3 e-ISBN978-1-84996-108-0
DOI10.1007/978-1-84996-108-0
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Preface
Thisbookpresentsnewresultsonapplicationsofgeometricalgebra.Thetimewhen
researchersandengineerswerestartingtorealizethepotentialofquaternionsforap-
plicationsinelectrical,mechanic,andcontrolengineeringpassedalongtimeago.
SincethepublicationofSpace-TimeAlgebrabyDavidHestenes(1966)andClifford
AlgebratoGeometricCalculus:AUnifiedLanguageforMathematicsandPhysics
by David Hestenes and Garret Sobczyk (1984), consistent progress in the appli-
cations of geometric algebra has taken place. Particularly due to the great devel-
opments in computer technology and the Internet, researchers have proposed new
ideasandalgorithmstotackleavarietyofproblemsintheareasofcomputerscience
andengineeringusingthepowerfullanguageofgeometricalgebra.Inthisprocess,
pioneer groups started the conference series entitled “Applications of Geometric
AlgebrainComputerScienceandEngineering”(AGACSE)inordertopromotethe
research activity in the domain of the application of geometric algebra. The first
conference, AGACSE’1999, organized by Eduardo Bayro-Corrochano and Garret
Sobczyk, took place in Ixtapa-Zihuatanejo, Mexico, in July 1999. The contribu-
tionswerepublishedinGeometricAlgebrawithApplicationsinScienceandEngi-
neering,Birkhäuser,2001.Thesecondconference,ACACSE’2001,washeldinthe
EngineeringDepartment of the CambridgeUniversity on 9–13 July 2001 and was
organizedbyLeoDorst,ChrisDoran,andJoanLasenby.Thebestconferencecontri-
butionsappearedasabookentitledApplicationsofGeometricAlgebrainComputer
ScienceandEngineering,Birkhäuser,2002.Thethirdconference,AGACSE’2008,
took place in August 2008 in Grimma, Leipzig, Germany. The conference chairs,
EduardoBayro-CorrochanoandGerikSheuermann,editedthisbookusingselected
contributionsthatwerepeer-reviewedbyatleasttworeviewers.
In the history of science, theories would have not been developed at all with-
outessentialmathematicalconcepts.Invariousperiodsofthehistoryofmathemat-
ics and physics, there is clear evidence of stagnation, and it is only thanks to new
mathematicaldevelopmentsthatastonishingprogresshastakenplace.Furthermore,
researchers unavoidably cause fragmented knowledge in their various attempts to
combine different mathematical systems. We realize that each mathematical sys-
tembringsaboutsomepartsofgeometry;however,together,theyconstituteasys-
tem that is highly redundant due to an unnecessary multiplicity of representations
v
vi Preface
forgeometricconcepts.Incontrast,inthegeometricalgebralanguage,mostofthe
standardmattertaughtinengineeringandcomputersciencecanbeadvantageously
reformulatedwithoutredundanciesandinahighlycondensedfashion.
Thisbookpresentsaselectionofarticlesaboutthetheoryandapplicationsofthe
advancedmathematicallanguagegeometricalgebrawhichgreatlyhelpstoexpress
theideasandconceptsandtodevelopalgorithmsinthebroaddomainsofcomputer
scienceandengineering.Thecontributionsareorganizedinsevenparts.
The first part presents screw theory in geometric algebra, the parameterization
of3Dconformaltransformationsinconformalgeometricalgebra,andanoverview
ofapplicationsofgeometricalgebra.Thesecondpartincludesthoroughstudieson
Cliffor–Fourier transforms: the two-dimensional Clifford windowed Fourier trans-
form; the cylindrical Fourier transform; applications of the 3D geometric algebra
Fourier transform in graphics engineering; the 4D Clifford–Fourier transform for
color image processing; and the use of the Hilbert transforms in Clifford analysis
for signal processing. In the third part, self-organizing geometric neural networks
areutilizedfor2Dcontourand3Dsurfacereconstructioninmedicalimageprocess-
ing.Theclusteringandclassificationarehandledusinggeometricneuralnetworks
and associative memories designed in the conformal geometric algebra. This part
concludes with a retrospective of the quaternion wavelet transform, including an
applicationforstereovision.Thefourthpartforcomputervisionstartswithanew
cone-pixelcamera using a convexhull and twists in conformal geometric algebra.
Thenextworkintroducesamodel-basedapproachforglobalself-localizationusing
activestereovisionandGaussianspheres.Inthefifthpart,thegeometriccharacter-
izationofM-conformalmappingsisdiscussed,andastudyoffluidflowproblems
is carried out in depth using quaternionic analysis. The sixth part shows the im-
pressive space group visualizer for all 230 3D groups using the software packet
forgeometricalgebracomputationsCLUCalc.Thesecondauthorstudiesgeometric
algebraformalismasanalternativetodistributedrepresentationmodels;herecon-
volutions are replaced by geometric products, and, as a result, a natural language
for visualization of higher concepts is proposed. Another author studies computa-
tionalcomplexityreductionsusingCliffordalgebrasandshowsthatgraphproblems
of complexity class NP are polynomial in the number of Clifford operations re-
quired.Theseventhpartincludesnewdevelopmentsinefficientgeometricalgebra
computing: The first author presents an efficient blade factorization algorithm to
producefasterimplementationsoftheJoin;withthesoftwarepacketGALOOP,the
second author symbolically reduces involved formulas of conformal geometric al-
gebra,generatingsuitablecodeforcomputingusinghardwareaccelerators.Another
chaptershowsapplicationsofGrobnerbasesinrobotics,formulatedinthelanguage
of Clifford algebras, in engineering to the theory of curves, including Fermat and
Beziercubics,andintheinterpolationoffunctionsusedinfiniteelementtheory.
We are very thankful to all book contributors, who are working persistently to
advancetheapplicationsofgeometricalgebra.Wedohopethatthereaderwillfind
this collection of contributions in a broad scope of the areas of engineering and
computer science very stimulating and encouraging. We hope that, as a result, we
willseeourcommunitygrowingandbenefittingfromnewandpromisingscientific
Preface vii
contributions. Finally, we thank also for the support to this book project given by
CINVESTAVUnidadGuadalajaraandCONACYTProject2007-182084.
CINVESTAV,Guadalajara,México EduardoBayro-Corrochano
UniversitätLeipzig, GerikSheuermann
InstitutfürInformatik,Germany

Contents
PartI GeometricAlgebra
NewToolsforComputationalGeometryandRejuvenationofScrew
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
DavidHestenes
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 UniversalGeometricAlgebra . . . . . . . . . . . . . . . . . . . . 4
3 GroupTheorywithGeometricAlgebra . . . . . . . . . . . . . . . 6
4 EuclideanGeometrywithConformalGA . . . . . . . . . . . . . . 8
5 InvariantEuclideanGeometry . . . . . . . . . . . . . . . . . . . . 10
6 ProjectiveGeometry . . . . . . . . . . . . . . . . . . . . . . . . . 13
7 CovariantEuclideanGeometrywithConformalSplits . . . . . . . 14
8 RigidDisplacements . . . . . . . . . . . . . . . . . . . . . . . . . 18
9 FramingaRigidBody . . . . . . . . . . . . . . . . . . . . . . . . 20
10 RigidBodyKinematics . . . . . . . . . . . . . . . . . . . . . . . 22
11 RigidBodyDynamics . . . . . . . . . . . . . . . . . . . . . . . . 24
12 ScrewTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
13 ConformalSplitandMatrixRepresentation . . . . . . . . . . . . . 28
14 LinkedRigidBodies&Robotics . . . . . . . . . . . . . . . . . . 31
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Tutorial:Structure-PreservingRepresentationofEuclideanMotions
ThroughConformalGeometricAlgebra . . . . . . . . . . . . . . . . 35
LeoDorst
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 ConformalGeometricAlgebra . . . . . . . . . . . . . . . . . . . 36
2.1 Trick1:RepresentingEuclideanPointsinMinkowskiSpace 36
2.2 Trick 2: Orthogonal Transformations as Multiple
ReflectionsinaSandwichingRepresentation . . . . . . . . 39
2.3 Trick3:ConstructingElementsbyAnti-Symmetry . . . . . 42
2.4 Trick4:DualSpecificationofElementsPermitsIntersection 43
ix