Exponentials and motions in geometric algebra

L Simpson, Glen Mullineux

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

Abstract

The use of geometric algebra to define and manipulate rigid-body motions is investigated. An algebra with four basis elements of grade 1 is used in which the square of one of these elements is regarded as being infinite. This gives a representation of projective space and allows rotations and translations to be defined exactly. By smoothly interpolating between such transforms, smooth motions can be created using techniques such as spherical linear interpolation (Slerp). This requires the ability to handle the exponential function within the algebra. A closed form expression for the exponential is derived in the general case when the square of the special basis element is any real number. Taking this to be infinite allows smooth motions to be created and some examples are presented.
Original languageEnglish
Title of host publicationGraVisMa 2009: Workshop Proceedings
Subtitle of host publicationInternational Workshop on Computer Graphics, Computer Vision and Mathematics in co-operation with EUROGRAPHICS
EditorsV Skala, D Hildenbrand
Place of PublicationPlzen, CZ
PublisherVaclav Skala/ University of West Bohemia
Pages9-16
Number of pages8
ISBN (Print)9788086943909
Publication statusPublished - 2009
EventInternational Workshop on Computer Graphics, Computer Vision and Mathematics (GraVisMa) 2009 - University of West Bohemia, Plzen, Czech Republic
Duration: 1 Sep 2009 → …

Conference

ConferenceInternational Workshop on Computer Graphics, Computer Vision and Mathematics (GraVisMa) 2009
CountryCzech Republic
CityUniversity of West Bohemia, Plzen
Period1/09/09 → …

Fingerprint

Geometric Algebra
Rigid Body Motion
Algebra
Linear Interpolation
Motion
Projective Space
Closed-form
Transform

Cite this

Simpson, L., & Mullineux, G. (2009). Exponentials and motions in geometric algebra. In V. Skala, & D. Hildenbrand (Eds.), GraVisMa 2009: Workshop Proceedings: International Workshop on Computer Graphics, Computer Vision and Mathematics in co-operation with EUROGRAPHICS (pp. 9-16). Plzen, CZ: Vaclav Skala/ University of West Bohemia.

Exponentials and motions in geometric algebra. / Simpson, L; Mullineux, Glen.

GraVisMa 2009: Workshop Proceedings: International Workshop on Computer Graphics, Computer Vision and Mathematics in co-operation with EUROGRAPHICS. ed. / V Skala; D Hildenbrand. Plzen, CZ : Vaclav Skala/ University of West Bohemia, 2009. p. 9-16.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Simpson, L & Mullineux, G 2009, Exponentials and motions in geometric algebra. in V Skala & D Hildenbrand (eds), GraVisMa 2009: Workshop Proceedings: International Workshop on Computer Graphics, Computer Vision and Mathematics in co-operation with EUROGRAPHICS. Vaclav Skala/ University of West Bohemia, Plzen, CZ, pp. 9-16, International Workshop on Computer Graphics, Computer Vision and Mathematics (GraVisMa) 2009, University of West Bohemia, Plzen, Czech Republic, 1/09/09.
Simpson L, Mullineux G. Exponentials and motions in geometric algebra. In Skala V, Hildenbrand D, editors, GraVisMa 2009: Workshop Proceedings: International Workshop on Computer Graphics, Computer Vision and Mathematics in co-operation with EUROGRAPHICS. Plzen, CZ: Vaclav Skala/ University of West Bohemia. 2009. p. 9-16
Simpson, L ; Mullineux, Glen. / Exponentials and motions in geometric algebra. GraVisMa 2009: Workshop Proceedings: International Workshop on Computer Graphics, Computer Vision and Mathematics in co-operation with EUROGRAPHICS. editor / V Skala ; D Hildenbrand. Plzen, CZ : Vaclav Skala/ University of West Bohemia, 2009. pp. 9-16
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