Geometric algebra (based on Clifford algebra)

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 Clifford algebra. 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 to it.

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.

• FREQUENTLY ASKED QUESTIONS

• SOFTWARE FOR GEOMETRIC ALGEBRA
  
  • A more comprehensive listing of available software and utilities may be found at http://sinai.mech.fukui-u.ac.jp/gcj/gc_int .
  • GAIGEN, a C++ code optimized generator
    
  • GAVIEWER, an interactive viewer (including the conformal model) based on GAIGEN.
    
  • CLU, a C++ library with OPEN-GL graphics
    
  • GABLE, a Matlab package with visualization
    
  • 'the Cambridge group software', a Maple V package
    
  • CLIFFORD for Maple V
  • CLICAL, a Clifford algebra calculator

• 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).
    The paper Modelling 3D Euclidean geometry by D. Fontijne and L.Dorst can be viewed as part III (CGA March/April 2003).

  • GABLE (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 GAVIEWER site.

  • SIGGRAPH01-presentation. A mockup of the illustrated and animated presentation Steve Mann 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 David Hestenes also contains many introductions, at various levels.

  • Cinderella illustrations: I am making JAVA applets illustrating various 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.

  • A paper 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.) PDF here.

  • 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 ps or pdf.
    A related presentation is on the systems theory of contact, in ps or pdf.
    

  

• LINKS
  
  • Hermann GünterGrassmann, William Kingdon Clifford
  • Lots of paper on the The University of Cambridge groups at astronomy and engineering.
  • The Kiel group (Sommer, Perwass, Rosenhahn et al.)
  • Jaap Suter (the Clifford software, and a tutorial)
  • Pertti Lounesto's page - after Pertti's sudden death, there is fortunately a mirror of his site.
  • Ian Bell (concise introduction to multivector methods)
  • For computer graphics: Steve Mann (introductory talks, GABLE info)