Geomerics - Advancing Technology Through Innovation

Geometric algebra (GA) is an advanced mathematical technique that has been developed and refined for over 100 years. It was developed by the noted Cambridge mathematician and scientist W.K. Clifford in the late 19^th Century and since then has spread to many areas of mathematics, physics and engineering.

Geometric algebra has been shown to dramatically simplfy calculations in a range of physical and geometric problems. It also acts as a higher level mathematical language, making it easier to transport innovations from one area into seemingly unrelated applications.

GA provides a full algebra over the vectors (including novel concepts such as division by a vector). This allows for a number of geometic primitive to be encoded in a way that preserves all the information about them. It then becomes possible, for example, to divide by a circle, or multiply by a sphere, and the results of such operations are always geometrically meaningful. Similarly operations such as rotations, translations and reflections are also encoded directly as elements of the algebra.

These algebraic basics provide the obvious starting point for a range of problems in geometry and physics. Collision detection and rigid-body dynamics fall out neatly (the latter using a generalisation of the quaternion algebra), as do problems in lighting, ray-tracing and motion capture.

Resources

If you are new to the subject, we recommend the following introductory resources:

• Jaap Suter's GA primer.
• The course notes from SIGGRAPH 2000 and SIGGRAPH 2001.
• "Using Geometric Algebra for Computer Graphics" in Games Programming Gems 5, p. 201.
• " Imaginary Numbers are Not Real" by S. Gull, A. Lasenby and C. Doran. This is more suitable for those with some physics knowledge.
• " Geometric Algebra for Physicists" by C. Doran and A. Lasenby. The most complete reference on geometric algebra, written assuming no prior knowledge of GA, but illustrates the key themes predominantly with examples from physics.
• " Conformal Geometry, Euclidean Space and Geometric Algebra", by Chris Doran, Anthony Lasenby and Joan Lasenby

Research Groups

There are a number of research groups dedicated to GA around the world. These are the main ones (with due apologies to anyone left off the list).

• The Cambridge University Geometric Algebra Research Group Homepage.
• David Hestenes' Geometric Calculus Home Page
• International Clifford Algebra Society (formerly the Clifford Algebra Bulletin Board). Their abtracts and preprints are also available by ftp.
• Clifford Research Group - University of Gent, Belgium
• Leo Dorst - University of Amsterdam
• Robin Tucker - Lancaster University
• Christian Perwass - Kiel University. Includes links to his software library.
• Ian Bell's introduction to GA. (Yes, that is the same Ian Bell of Elite fame.)
• Jaap Suter's GA primer
• Geometric Calculus in Japan

Software

A number of groups also distribute free software to get you going with GA. The following are particularly useful:

• The Cambridge University GA Group downloads page, including routines for Maple and Mathematica.
• A Clifford Algebra package available for Maple from Rafal Ablamowicz at the Maths Department of Gannon University, PA, USA.
• Software to accompany the book "Clifford Algebras with Numeric and Symbolic Computations" edited by Rafal Ablamowicz, Pertti Lounesto and Josep Parra is avalable from the Birkhäuser Boston web site.