Computational synthetic geometry with Clifford algebra

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.