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.
|Title of host publication||GraVisMa 2009: Workshop Proceedings|
|Subtitle of host publication||International Workshop on Computer Graphics, Computer Vision and Mathematics in co-operation with EUROGRAPHICS|
|Editors||V Skala, D Hildenbrand|
|Place of Publication||Plzen, CZ|
|Publisher||Vaclav Skala/ University of West Bohemia|
|Number of pages||8|
|Publication status||Published - 2009|
|Event||International Workshop on Computer Graphics, Computer Vision and Mathematics (GraVisMa) 2009 - University of West Bohemia, Plzen, Czech Republic|
Duration: 1 Sep 2009 → …
|Conference||International Workshop on Computer Graphics, Computer Vision and Mathematics (GraVisMa) 2009|
|City||University of West Bohemia, Plzen|
|Period||1/09/09 → …|
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). Vaclav Skala/ University of West Bohemia.