Graphics Object-Oriented Platform for Euclidean Geometry Computations

Abstract

Recent object-oriented programming languages are enabling the top level code in application programs to resemble more closely the form of the mathematical expressions that the program is meant to be implementing. This facility is very useful for non-programmers, and mathematicians and geometers who are not interested in the fine syntactic details of computer programming languages. This paper describes an object-oriented platform that makes it easier for non-professional programmers to implement and test concepts from standard Euclidean geometry on a computer graphics screen. The idea is that this platform enables one to construct and test geometric hypotheses and theorems in a language closely resembling the way Euclid and traditional geometry expresses geometric concepts, symbols and theorems. Although the language used by Euclid for geometry is precise it also includes the contextual facilities of natural languages saveing one from having to spell out every characteristic and attribute in detail. It is this demand for completeness in specifying details that has made standard computer programming languages laborious and tedious to deal with. The graphics object-oriented platform described in this paper incorporates the facility for handling incompleteness in a natural and visually acceptable way. Additionally the platform incorporates constraint resolution by an improved iteration technique. Finally, the platform contains the hierarchy of geometrical shapes to which the geometer needs immediate access. Here it is pointed out that the object-oriented programming object hierarchy is properly the inverse of the conceptual geometrical hierarchy of shapes.