Geometric algebra (based on Clifford algebra)
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Geometric algebra is a very convenient representational and computational
system for geometry. It is going to be
the way computer science deals with geometrical issues. It contains,
in a fully integrated manner, linear algebra, vector calculus, differential
geometry, complex numbers and quaternions as real geometric entities,
and lots more. This powerful language is based in
David Hestenes was the among first to realize its enormous importance
for physics, where it is now finding inroads. The revolution for
computer science is currently in the making, and we hope to contribute
WE ARE IN THE PROCESS OF RE-ORGANIZING OUR AMSTERDAM GEOMETRIC ALGEBRA PAGES,
AND RELOCATING THEM TO
http://www.science.uva.nl/ga/. PLEASE USE THAT LINK IN THE FUTURE.
last update: 20020608
FREQUENTLY ASKED QUESTIONS
SOFTWARE FOR GEOMETRIC ALGEBRA
INTRODUCTIONS TO GEOMETRIC ALGEBRA
- The book Geometric Algebra for Computer Science: an object oriented approach to geometry, by L.Dorst, D.Fontijne and S.Mann, Morgan Kaufman publishers 2007.
- The book Geometric Algebra for Physicists, by C.Doran and A.Lasenby, Cambridge University Press, 2003, 2007.
- Geometric algebra: a computational framework for geometrical
applications, by L.Dorst and S.Mann.
A two-part paper in
IEEE Computer Graphics and Applications in 2002:
part I (May/June 2002)
part II (July/August 2002).
Modelling 3D Euclidean geometry by D. Fontijne and L.Dorst can be
part III (CGA March/April 2003).
(Geometric AlgeBra Learning Environment,
by Leo Dorst, Steve Mann and Tim Bouma)
is a tutorial written in Matlab; lots of visualization, and a (hopefully)
generally accessible tutorial text to accompany it.
An motivational introductory talk on the subject:
Geometric (Clifford) algebra: a practical tool for efficient geometric
representation (Leo Dorst, 1999). Renewed 5/99
(pdf version here)
Another talk, from fast introduction to a ray tracer implementation:
Geometric algebra, the framework for geometic computations
(Leo Dorst, 2002)
Presentation Exploring the conformal model
of 3D Euclidean geometry
given at ITMga2003, Kyoto, Japan, november 2003.
(PDF 600k, looks crappy on screen but prints well.)
The interactive demos
which this presentation refers to (and which generated the pictures)
will be made available on the
A mockup of the illustrated and animated presentation
and I gave at SIGGRAPH 2001.
Tutorial material: besides
GABLE, there is
Physical Applications of Geometric Algebra by Doran and Lasenby.
For engineering/computer vision, there is the introductory paper
New Geometric Methods for Computer Vision by Joan Lasenby et al.
The site of
also contains many introductions, at various levels.
Cinderella illustrations: I am making
geometrical concepts and constructions in geometric algebra.
PAPERS AND PRESENTATIONS
My paper The inner products of geometric
algebra which appeared in the book
Applications of Geometric Algebra in Computer Science and Engineering
(Dorst, Doran, Lasenby, eds), Birkhauser, 2002.
Honing geometric algebra
for its use in the computer sciences
(Leo Dorst, 2001) published in the book
Geometric Computing with Clifford Algebras,
ed. G. Sommer, Springer 2001, Chapter 6, pp. 127-152.
(The book version lacks some symbols in the figures.)
- A paper on an application:
Objects in contact: boundary collisions as geometric wave propagation
(Leo Dorst, 2001), in
E. Bayro-Corrochano, G. Sobczyk, eds, Birkh\"auser, 2001,
Chapter 17, pp. 349-370.
PDF here (looks ugly, prints well).
Also available: the slides of this presentation, in
A related presentation is on the systems theory of contact,