Solving geometrical constraint systems using CLP based on linear constraint solver

Abstract

Euclidean geometrical configurations obtained with ruler, square and compass may be described as arithmetic constraint systems over rational numbers and consequently belong to the domain of CLP(R). Unfortunately, CLP based on linear constraint solvers which are efficient and can deal with geometrical constraints such as parallelism, perpendicularity, belonging to a line i.e. pseudo-linear constraints, cannot handle quadratic constraints introduced when using circles.Two problems arise with quadratic constraints: the first problem is how to solve mixed constraint systems i.e. linear constraints combined with quadratic constraints; the second problem is how to represent the real numbers involved in the resolution of mixed constraints, so that correctness and completeness of linear constraint solvers are preserved.In this paper we present a naive algorithm for mixed constraints based on a cooperation with a linear constraint solver. We define a representation for the real numbers, i.e. constructible numbers, occuring in Euclidean geometry. This representation preserves correctness and completeness of above algorithms. A survey over 512 theorems of Euclidean geometry shows that from both theoretical and experimental points of view, this representation is appropriate. This work is intended to be used to verify geometrical properties in Intelligent Tutoring System for geometry.Keywordsquadratic algebraic extensionrepresentation of rational, real, algebraic, and constructible numberscooperation between solvers for mixed constraints