A dynamic symbolic geometry environment based on the Gr\"obnerCover algorithm for the computation of geometric loci and envelopes

TL;DR

This paper presents an enhanced dynamic geometry system that uses the Gr"obnerCover algorithm to automatically compute algebraic loci and envelopes from geometric constructions, demonstrating practical applicability.

Contribution

It introduces a prototype that integrates the Gr"obnerCover algorithm into GeoGebra for symbolic locus and envelope computation, advancing dynamic geometry software capabilities.

Findings

Successfully computes algebraic loci and envelopes from constructions

Demonstrates applicability in general purpose dynamic geometry systems

Enhances GeoGebra with symbolic algebraic computation features

Abstract

An enhancement of the dynamic geometry system GeoGebra for the automatic symbolic computation of algebraic loci and envelopes is presented. Given a GeoGebra construction, the prototype, after rewriting the construction as a polynomial system in terms of variables and parameters, uses an implementation of the recent Gr\"obnerCover algorithm to obtain the algebraic description of the sought locus/envelope as a locally closed set. The prototype shows the applicability of these techniques in general purpose dynamic geometry systems.