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Computational synthetic geometry with Clifford algebra

  • Timothy F. Havel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1360)

Abstract

Computational synthetic geometry is an approach to solving geometric problems on a computer, in which the quantities appearing in the equations are all covariant under the corresponding group of transformations, and hence possess intrinsic geometric meanings. The natural covariant algebra of metric vector spaces is called Clifford algebra, and it includes Gibbs' vector algebra as a special case. As a preliminary essay demonstrating that one can develop practical computer programs based on this approach for solving problems in Euclidean geometry, we have implemented a MAPLE package, called Gibbs, for the elementary expansion and simplification of expressions in Gibbs' abstract vector algebra. We also show how to translate any origin-independent scalar-valued expression in the algebra into an element of the corresponding invariant ring, which we have christened the Cayley-Menger ring. Finally, we illustrate the overall approach by using it to derive a new kinematic parametrization of the conformation space of the molecule cyclohexane.

Keywords

Euclidean Geometry Clifford Algebra Geometric Algebra Triple Product Chair Conformation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Timothy F. Havel
    • 1
  1. 1.Biological Chemistry and Molecular PharmacologyHarvard Medical SchoolBoston

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