Symbolic Geometric Reasoning with Advanced Invariant Algebras

Abstract

In symbolic geometric reasoning, the output of an algebraic method is expected to be geometrically interpretable, and the size of the middle steps is expected to be sufficiently small for computational efficiency. Invariant algebras often perform well in meeting the two expectations for relatively simple geometric problems. For example in classical geometry, symbolic manipulations based on basic invariants such as squared distances, areas and volumes often have great performance in generating readable proofs. For more complicated geometric problems, the basic invariants are still insufficient and may not generate geometrically meaningful results.

An advanced invariant is a monomial in an “advanced algebra”, and can be expanded into a polynomial of basic invariants that are also included in the algebra. In projective incidence geometry, Grassmann-Cayley algebra and Cayley bracket algebra are an advanced algebra in which the basic invariants are determinants of homogeneous coordinates of points, and the advanced invariants are Cayley brackets. In Euclidean conformal geometry, Conformal Geometric Algebra and null bracket algebra are an advanced algebra where the basic invariants are squared distances between points and and signed volumes of simplexes, and the advanced invariants are Clifford brackets.

This paper introduces the above advanced invariant algebras together with their applications in automated geometric theorem proving. These algebras are capable of generating extremely short and readable proofs. For projective incidence theorems, the proofs generated are usually two-termed in that the conclusion expression maintains two-termed during symbolic manipulations. For Euclidean geometry, the proofs generated are mostly one-termed or two-termed.