A parametric approach to 3D dynamic geometry

Dynamic geometry systems are computer applications allowing the exact on– screen drawing of geometric diagrams and their interactive manipulation by mouse dragging. Whereas there exists an extensive list of 2D dynamic geometry environments, the number of 3D systems is reduced. Most of them, both in 2D and 3D, share a common approach for the management of geometric constructions, using numerical data for the present diagram instance and elementary methods for computing derived objects. This paper deals with a parametric approach for automatic management of 3D Euclidean constructions. An open source library, implementing the core functions in a 3D dynamic geometry system, is described here. The library deals with constructions by using symbolic parameters, so allowing a full algebraic knowledge about objects such as loci and envelopes. This parametric approach is also a prerequisite for performing automatic proof, and basic functions are defined for symbolically checking the truth of statements. Using recent results coming from the theory of parametric polynomial systems solving, the bottleneck in the automatic determination of geometric loci and envelopes is solved. As far as we know, there is no comparable library in the 3D case, except the paramGeo3D library (designed for computing equations of simple 3D geometric objects, but it lacks specific functions for finding loci and envelopes).