A parametric approach to 3D dynamic geometry

Francisco Botana

doi:10.1016/j.matcom.2012.12.004

Abstract

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, using numerical data to manage geometric knowledge and elementary methods to compute 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, thus enabling a full algebraic knowledge about objects such as loci and envelopes. This parametric approach is also a prerequisite for performing automatic proof. Basic functions are defined for symbolically checking the truth of statements. Using recent results 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, which, however, lacks specific functions for finding loci and envelopes).

Highlights

A method for mechanical theorem proving in geometries was proposed by Wu [28, 29] during the eighties, and, almost simultaneously, a novel applica-Preprint submitted to Elsevier tion for the algorithm of Buchberger [8], related with automated deduction, was studied
These programs marked the birth of the dynamic geometry (DG) paradigm, and formed the core of mathematical sofwtare used in schools
Botana and Valcarce [1, 3] develop a DG environment that communicates with CoCoA [10] and Mathematica, and where Grobner bases are used for loci finding and automated proof and discovery

Summary

Introduction

A method for mechanical theorem proving in geometries was proposed by Wu [28, 29] during the eighties, and, almost simultaneously, a novel applica-. Preprint submitted to Elsevier tion for the algorithm of Buchberger [8], related with automated deduction, was studied (see, for instance, [15]) Ending this decade two learning environments for geometry appeared in the educational software market, The Geometer’s Sketchpad, GSP, [38] and Cabri [32]. Roanes–Lozano et al [21] use GSP constructions to perform proving tasks by means of a Maple library with a simple implementation of Wu’s method.

Issues in 2D

State of the art in 3D

Rationale of the library

Envelopes

Refining the algebraic descriptions of loci and envelopes

Conclusion