Guide to Geometric Algebra in Practice pp 435-453 | Cite as

# Tutorial Appendix: Structure Preserving Representation of Euclidean Motions Through Conformal Geometric Algebra

## Abstract

Using conformal geometric algebra, Euclidean motions in *n*-D are represented as orthogonal transformations of a representational space of two extra dimensions, and a well-chosen metric. Orthogonal transformations are representable as multiple reflections, and by means of the geometric product this takes an efficient and structure preserving form as a ‘sandwiching product’. The antisymmetric part of the geometric product produces a spanning operation that permits the construction of lines, planes, spheres and tangents from vectors, and since the sandwiching operation distributes over this construction, ‘objects’ are fully integrated with ‘motions’. Duality and the logarithms complete the computational techniques.

The resulting geometric algebra incorporates general conformal transformations and can be implemented to run almost as efficiently as classical homogeneous coordinates. It thus becomes a high-level programming language which naturally integrates quantitative computation with the automatic administration of geometric data structures.

This appendix provides a concise introduction to these ideas and techniques. *Editorial note*: This appendix is a slightly improved version of (Dorst in: Bayro-Corrochano, E., Scheuermann, G. (eds.) Geometric Algebra Computing for Engineering and Computer Science, pp. 457–476, [2011]). We provide it to make this book more self-contained.

## Keywords

Extra Dimension Direction Vector Rigid Body Motion Orthogonal Transformation Geometric Algebra## References

- 1.Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2000) Google Scholar
- 2.Dorst, L.: Tutorial: Structure preserving representation of Euclidean motions through conformal geometric algebra. In: Bayro-Corrochano, E., Scheuermann, G. (eds.) Geometric Algebra Computing for Engineering and Computer Science, pp. 457–476. Springer, Berlin (2011) Google Scholar
- 3.Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry. Morgan Kaufman, San Mateo (2007/2009). See www.geometricalgebra.net Google Scholar
- 4.Fontijne, D.: Efficient Implementation of Geometric Algebra. University of Amsterdam, Amsterdam (2007), ISBN: 13 978-90-889-10-142, online at www.science.uva.nl/~fontijne Google Scholar
- 5.Hestenes, D., Rockwood, A., Li, H.: System for encoding and manipulating models of objects. U.S. Patent 6,853,964, February 8, 2005 Google Scholar