Geometric Reasoning with Invariant Algebras

Abstract

Geometric reasoning is a common task in Mathematics Education, Computer-Aided Design, Computer Vision and Robot Navigation. Traditional geometric reasoning follows either a logical approach in Artificial Intelligence, or a coordinate approach in Computer Algebra, or an approach of basic geometric invariants such as areas, volumes and distances. In algebraic approaches to geometric reasoning, geometric interpretation is needed for the result after algebraic manipulation, but in general this is a difficult task. It is hoped that more advanced geometric invariants can make some contribution to the problem.

In this paper, a hierarchical framework of invariant algebras is introduced for hierarchical algebraic manipulation of geometric problems. The bottom level is the algebra of basic geometric invariants, and the higher levels are more complicated invariants. High level invariants can keep more geometric nature within algebraic structure. They are beneficial to geometric explanation but more difficult to handle than low level ones. The research focuses on doing geometric computation at high invariant levels.

The rest of this paper introduces algebraic techniques to manipulate high-level invariants and employs them in geometric reasoning, including both automated theorem proving and new theorem discovering. Using hierarchical invariant algebras one can obtain better geometric result from algebraic computation, in addition to more efficient algebraic manipulation.