Design And Implementation Of A Geometric-Object-Oriented Language

This thesis presents the design and implementation of a geometric-object-oriented language for symbolic geometric computation, reasoning, and visualization. This language allows users to construct geometric objects in plane geometry or other geometry spaces with certain or uncertain data. In this language, users can easily modify the constructed objects such as by renaming variables and changing values of parameters, effectively perform operations and calculations on them, and declare relations among them and formulate such relations with a proper structure. With the constructed objects and their relations, geometric problems can be formulated with specified syntax, and then they can be verified or proved with advanced techniques. A system implemented on the basis of this language will allow the user to perform geometric computation and reasoning rigorously, to prove and discover geometric theorems, and generate static or dynamic geometric diagrams and interactive documents automatically.In this language, we adopt a case distinction technique for formalizing symbolic geometric objects and relations. This technique allows users to construct and manipulate geometric objects and perform geometric computations and reasoning by enumerating all possible distinct cases. Based on the technique, two important concepts, composed geometric objects and composed geometric relations, are introduced. The case distinction technique reduces the problem of handling uncertainty and degeneracy to that of manipulating complicated side conditions involved in composed geometric objects and relations, and then to the problem of constraint handling. In this thesis, we introduce some special techniques and strategics for geometric constraint handling: representation, simplification, consistency checking, and relation derivation.This thesis mainly discusses the design and implementation of the proposed language. It first gives a survey on geometric computation and reasoning, in particular the development of geometric theorem proving, and reviews some of the important methods and software packages. Then the thesis de- tails the formalism of case distinction by defining composed geometric objects and relations, presents the expected capabilities and overall design of this language, and its main components and functionalities of each component. Some implementation issues are addressed based on the design including selection of programming language, models of implementation, and some main data structure. After that, some problems about constraints handling are discussed, and a specialized approach is introduced to solve geometric constraints involving algebraic inequalities dynamically and the process is presented to generate dynamic geometric diagrams automatically. The thesis is concluded with a summary and a brief discussion on some problems encountered but not completely settled such as managing geometric knowledge, solving semi-algebraic systems, and keeping the language more geometric.