Articulating Space

222 Pages·2016·29.61 MB·English

UNIVERSITY OF CALIFORNIA
Santa Barbara
Articulating Space:
Geometric Algebra for Parametric Design –
Symmetry, Kinematics, and Curvature
A Dissertation submitted in partial satisfaction
of the requirements for the degree of
Doctor of Philosophy
in
Media Arts and Technology
by
Pablo Colapinto
CommitteeinCharge:
ProfessorMarkoPeljhan,Chair
ProfessorJoAnnKuchera-Morin
ProfessorCurtisRoads
March2016
ThedissertationofPabloColapintoisapproved:
JoAnnKuchera-Morin
CurtisRoads
MarkoPeljhan,CommitteeChairperson
December2015
ArticulatingSpace: GeometricAlgebraforParametricDesign—
Symmetry,Kinematics,andCurvature
Copyright©2016
by
PabloColapinto
iii
Tofamilyandfriendsandmypartner,BrynDudkowski,
manythanksandlove.
iv
Acknowledgements
Withoutthediligentworkofothersthesepageswouldnotbepossible. Notably,theauthor
owesanincalculabledebttoProfessorLeoDorst,whosepublications,solutions,suggestions,
andhintshavebeenessentialindevelopingthecontainedformulations,andwhosethoughtful
commentsprovedvitaltothisdocument’srevision.
The PhD committee has provided critical insight, patient guidance, and comaraderie
throughout the preparation of this work. Professor Curtis Roads has been an active listener,
encouraging artistic exploration and contributing crucial context and terminology. Professor
JoAnnKuchera-Morinprovidedendlesssupport,trust,andanempoweringvisiontoadvance
our shared knowledge. The committee chair, Professor Marko Peljhan, offered a treasure
trove of intuition, analysis, and practical application in identifying the problem at hand and
ourcontributionstoit.
AttentivereadingsofanddiscussionswithJoanLasenbyandEduardoBayro-Corrochano
have proven quite formative as well, and the author would like to extend his thanks to them,
asofcoursetoDavidHestenesforencouragingusallto“domore.” Agraciousthanksisdue
Pierre Dechant, who provided essential training in the basics of reflection groups. Indeed,
many members of the geometric algebra community, including Garret Sobczyk, Dietmar
Hildenbrand,andIanBell,havedemonstratedanopennessinsharingtheirexpertknowledge.
In addition, this document has directly benefitted from communications with colleagues
both at MAT and beyond, including Wesley Smith, who has helped brainstorm and push the
GA way, as well as Karl Yerkes, Sölen Kiratli, Peter Flaherty, Charlie Roberts, Lance Put-
nam, Graham Wakefield, Matthew Wright, Amichi Amar, F. Myles Sciotto, Angus Forbes,
AugustBlack,StephenPope,theAlloSphereresearchgroup,ProfessorGeorgeLegrady,Pro-
fessor Marcos Novak, Professor Bhaskar Sarkar, Professor Rita Raley, and Professor Alfred
Guzzetti.
ResearchhasbeenfundedinpartthroughaRobertW.DeutschFoundationFellowshipvia
the AlloSphere Research Facility of the Media Arts and Technology Program at University
ofCaliforniaSantaBarbara,andbyanOliviaLongConverseFellowshipgrantedbyUCSB’s
GraduateDivision,withadditionalsupportprovidedbyTheSystemicsLabatMAT.Fortheir
financialsupport,theauthorwouldliketothanktheAlloSpheredirectorDr. JoAnnKuchera-
Morin,theDeutschFoundationPresidentJaneBrown,andtheSystemicsLabdirectorMarko
Peljhan, as well as the Graduate Division at UCSB and the Media Arts and Technology Pro-
gramadministrativestaff.
Ofcourse,anyerrorsoromissionsaretobeattributedtotheauthoralone.
v
Curriculum Vitae of Pablo Colapinto
December 2015
EDUCATION
UniversityofCalifornia,SantaBarbara 2007-2015
PhDinMediaArts&Technology,December2015.
MScinMediaArts&Technology,February2011.
HarvardUniversity 1996-2000
BAinVisualandEnvironmentalStudies,MagnaCumLaudewithHonors.
MellonFellow1998-2000;DeturBookAward1997.
AWARDS
RobertW.DeutschFoundationFellowship 2012-2015
OliviaLongConverseFellowship 2009-2012
PewFellowshipintheArts 2005-2006
PUBLICATIONS
Colapinto, P., Composing Surfaces with Conformal Rotors, In Proceedings of Applica-
tionsofGeometricAlgebrainComputerScienceandEngineering,Barcelona,2015.
Colapinto,P.andPeljhan,M.,GeometricAlgebraforDeployableDesign,InProceedings
of the International Association of Shell and Spatial Structures: Future Visions, Amster-
dam,2015.
Colapinto, P., Boosted Surfaces: Synthesis of 3D Meshes using Point Pair Generators
as Curvature Operators in the Conformal Model, Advances in Applied Clifford Algebras,
Volume24,Issue1,pp71-78,SpringerBasel,2013.
Colapinto, P., Versor: Spatial Computing with Conformal Geometric Algebra, Thesis
submitted in partial fulfillment of the requirements for the Master of Science degree in
MediaArtsandTechnology,UniversityofCalifornia,SantaBarbara,2011.
PRESENTATIONS
Functional Geometry: Producing Pure Spaces, C++ Now Conference, Aspen, Colorado,
2015.
vi
GenericProgrammingofGenericSpaces: Compile-timeGeometricAlgebrawithC++11,
C++NowConference,Aspen,Colorado,2014.
UpandRunningwithOpenGL.Tutorial,lynda.com,Carpenteria,CA,2014.
Laberintorios: Espaciossintéticosylageneracióndemovimientosenálgebrayagua,El
EncuentroTransmediayNarrativasAudiovisuales,Bogotá,2013.
ElGrupodelosTrece,PanelonLatinAmericaandCybernetics,InternationalSymposium
forElectronicArts,Albuquerque,2012.
OrganicFormsThroughTwistsandBoosts,InternationalSymposiumforElectronicArts,
Istanbul. 2011.
Informaciónes Imaginarias: Arte, Tecnología y la Manipulación de la Realidad, Festival
InternacionaldelaImagen,Manizales,2008.
RESEARCHANDTEACHINGEXPERIENCE
AlloSphereResearchFacility 2010-2015
GraduateResearchFellow SantaBarbara,CA
Collaboratewithresearcherstodevelopafully-immersive3Denvironmentforartisticand
·
scientificexploration.
UniversityofCalifornia 2007-2015
AssociateandAssistantTeaching SantaBarbara,CA
SoundandImage. TeachingAssociateandDeveloperofseriesofmultimediaengineering
·
coursesforCollegeofCreativeStudies.
Contemporary Cultural Theory and Introduction to Film Theory. Teaching Assistant,
·
FilmandMediaDepartment. Fall2008andWinter2009.
Experiments in Optical Computational Techniques. Teaching Assistant, Media Arts and
·
TechnologyProgram. Spring2008.
TempleUniversity/MooreCollegeofArtandDesign Spring2002-2007
AdjunctProfessor Philadelphia,PA
DevelopedcourseseriesonMediaArtsProgrammingfordepartmentofPhotographyand
·
DigitalArts,MooreCollegeofArtandDesign. Philadelphia,PA.
Innovated intermediate level courses on the theory and practice of non-linear video edit-
·
ing in the department of Broadcasting, Telecommunications and Mass Media and the
departmentofFilmandMediaArtsatTempleUniversity. Philadelphia,PA.
vii
ARTISTICPRODUCTION
MetropolitanOpera 2011-2013
VisualEffectsPost-ProductionSupervisor NewYorkCity,NY–Lyon,France
Directed a team of computer graphics specialists, 3D artists, and software engineers for
·
large-scaleprojectionintegratedintoproductionofWagner’sParsifal.
InstituteofContemporaryArt 2007
MediaArtist Philadelphia,PA
Created Pass Back a Revolver, a projection-mapped multimedia installation collabora-
·
tion with Peter Flaherty, as part of the Locally Localized Gravity exhibit. Available at
wolftype.com.
LincolnCenterandJapanSociety 2004-2005
3DGraphicsSpecialist NewYorkCity
Produced 3D video effects for My Life as a Fairy Tale. Director: Chen Shi-Zheng, Com-
·
poser: StephinMeritt,Producer: LincolnCenterFestival.
Developed creative projections for Dogugaeshi, an innovative puppet theatre. Director:
·
BasilTwist,Producer: TandemOtter.
TECHNICALSKILLS
Computer C++forMultimediaEngineering,OpenGL,Lua,Python
Mathematics GeometricAlgebrasandSpatialComputing
Software 3DModellingandVideoEditing
Languages English,Spanish,French
viii
Abstract
ArticulatingSpace: GeometricAlgebraforParametricDesign–Symmetry,Kinematics,and
Curvature
by
PabloColapinto
To advance the use of geometric algebra in practice, we develop computational methods
for parameterizing spatial structures with the conformal model. Three discrete parameteri-
zations – symmetric, kinematic, and curvilinear – are employed to generate space groups,
linkage mechanisms, and rationalized surfaces. In the process we illustrate techniques that
directly benefit from the underlying mathematics, and demonstrate how they might be ap-
plied to various scenarios. Each technique engages the versor – as opposed to matrix –
representation of transformations, which allows for structure-preserving operations on ge-
ometric primitives. This covariant methodology facilitates constructive design through geo-
metricreasoning: incidenceandmovementareexpressedintermsofspatialvariablessuchas
lines, circles and spheres. In addition to providing a toolset for generating forms and trans-
formations in computer graphics, the resulting expressions could be used in the design and
fabrication of machine parts, tensegrity systems, robot manipulators, deployable structures,
and freeform architectures. Building upon existing algorithms, these methods participate in
the advancement of geometric thinking, developing an intuitive spatial articulation that can
be creatively applied across disciplines, ranging from time-based media to mechanical and
structuralengineering,orreformulatedinhigherdimensions.
ix
Contents
1 Introduction: TransformingExpression 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 ProblemDefinition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 GeometricAlgebra 19
2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Beginnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 OrthogonalTransformations . . . . . . . . . . . . . . . . . . . . . . 20
2.2.2 CanonicalBasis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.3 BasisBlades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.4 GeometricProduct . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.5 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.6 TheMeet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.1 Versors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.2 Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.3 ProjectionandRejection . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.4 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.5 TheRemarkableExponential . . . . . . . . . . . . . . . . . . . . . . 32
2.3.6 RationalizingwithRatios . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.7 ExampleProblem: PreventingCameraRoll . . . . . . . . . . . . . . 36
2.3.8 OtherMetrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4 ConformalGeometricAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.1 ConformalMetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.2 Blades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.3 PrimitiveElements . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4.3.1 PointsasNullVectors . . . . . . . . . . . . . . . . . . . . 43
2.4.3.2 DirectRepresentation . . . . . . . . . . . . . . . . . . . . 46
2.4.3.3 DualRepresentation . . . . . . . . . . . . . . . . . . . . . 47
2.4.4 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
x