Reasoning of Geometric Concepts based on Algebraic Constraint-directed Method

We present an algebraic approach to geometric reasoning and learning. The purpose of this research is to avoid the usual difficulties in symbolic handling of geometric concepts. Our system GREW is grounded on a reasoning scheme that integrate the symbolic reasoning and algebraic reasoning of Wu's method. The basic principle of this scheme is to describe mathematical knowledge in terms of symbolic logic and to execute the subsidiary reasoning for Wu's method. The validity of our approach and GREW is shown by experiments, such as applying to learning-by-example of computer vision heuristics or solving locus problems. 1 Introduction This paper presents a new approach for learning or reasoning of geometric concepts based on algebraic constraint-directed methods. Geometric reasoning is available for many applications , such as robotics, CAD and computer vision. However , most previous reasoning systems, which are based on predicate logic, have difficulties in handling geometric notions. This is because the usual symbolic approach fails to grasp the essential characteristics of geometry, and cannot solve complicated problems, such as those which require auxiliary lines. As a result, handling geometric concepts causes great trouble in many applications of reasoning. For instance , consider the heuristics called skewed symmetry in computer vision [Kanade81]. This is a famous geometric constraint which claims that a two-dimensional skewed symmetry is a projected image of a genuine three-dimensional symmetry (Fig.1). Because of transformation-invariant characteristics such as shear transformation, it is very difficult to represent this constraint by usual predicate logic, still more to establish the reasoning system. In order to solve these difficulties , we select Wu's method as algebraic approach, and *I wish to thank members in FAI-WG (Foundation of Artificial Intelligence) and CLP-WG (Constraint Logic Programming) of ICOT for useful comments and discussion on earlier drafts of this work.