Rigid-Body Transforms Using Symbolic Infinitesimals
There is a requirement to be able to represent three-dimensional objects and their transforms in many applications, including computer graphics and mechanism and machine design. A geometric algebra is constructed which can model three-dimensional geometry and rigid-body transforms. The representation is exact since the square of one of the basis vectors is treated symbolically as being infinite. The non-zero, even-grade elements of the algebra represent precisely all rigid-body transforms. By allowing the transform to vary, smooth motions are obtained. This can be achieved using Bézier and B-spline combinations of even-grade elements.
KeywordsControl Point Basis Vector Geometric Algebra Curve Segment Smooth Motion
The work reported in the paper was carried within the Innovative Design and Manufacturing Research Centre at the University of Bath, and the second author is funded by a studentship provided via the Centre. The Centre is funded by the Engineering and Physical Sciences Research Council (EPSRC), and this support is gratefully acknowledged.
- 3.Farin, G.: Curves and Surfaces for CAGD: A Practical Guide, 5th edn. Morgan Kaufmann, San Francisco (2001) Google Scholar
- 7.González Calvet, R.: Treatise of Plane Geometry Through Geometric Algebra. Cerdanyola del Vallès (2007) Google Scholar
- 10.Jüttler, B., Wagner, M.G.: Computer-aided design with spatial rational B-spline motions. J. Mech. Des. 118, 193–201 (1996) Google Scholar
- 19.Simpson, L., Mullineux, G.: Exponentials and motions in geometric algebra. In: Vaclav, S., Hildenbrand, D. (eds.) International Workshop on Computer Graphics, Computer Vision and Mathematics (GraVisMa), pp. 9–16. Union Agency, Plzen (2009) Google Scholar