Computational Geometry

Geometric Algebra (GA) has been designed to support a rich correspondence between geometric constructions and algebraic structures. This makes it an ideal language for computer graphics and computational geometry. At the same time it improves computational efficiency with new, coordinate-free representations and methods.

As shown by many examples, GA has a broader range of applications to physics than any other mathematical language, including matrix algebra. This makes it a prime candidate for development as a universal system for scientific computations, with diverse applications to physics, engineering and computer science.

Recently a breakthrough of sorts has been acheived, which greatly simplifies representations and computations in Euclidean geometry. Of course this has ramifications and simplifications in every domain of applied science. The crucial role of GA is outlined in the first paper (Old Wine) below. As that paper is an invited talk for experts, it will be dificult for anyone unfamiliar with GA. A detailed treatment included in the subsequent set of four papers (A Unified Algebraic Framework), specifically in the second of the four. The first of the set is an introduction for neophytes in GA.

Quick entry into GA in robotics is provided by the two papers on Body Mechanics. The second of these has an appendix describing an advanced method, which has been significantly improved by the recent breakthrough, and explained in "Old Wine."

The last of the papers below (Symmetry Groups) applies GA to detailed treatment of the point groups in Euclidean 3-space. It was written about 20 years ago as a chapter for New Foundations for Mathematical Physics. A recent addition to include the space groups is included here.

  1. Old Wine in New Bottles: A new algebraic framework for computational geometry
  2. A Unified Algebraic Framework for Classical Geometry
    1.  New Algebraic Tools for Classical Geometry
  3. 2.  Generalized Homogeneous Coordinates for Computational Geometry
    3.  Spherical Conformal Geometry with Geometric Algebra
    4.  A Universal Model for Conformal Geometries of Euclidean, Spherical and Double-Hyperbolic Spaces
    5.  New Tools for Computational Geometry and rejuvenation of Screw Theory
  4. Invariant Body Kinematics
  5. I  Saccadic and Compensatory Eye Movements
    II  Reaching and Neurogeometry
    Homogeneous Rigid Body Mechanics with Elastic Coupling
  6. Symmetry Groups
  7. Point Groups and Space Groups in Geometric Algebra
    The Crystallographic Space Groups in Geometric Algebra

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