Software using the Gröbner Cover for geometrical loci computation and classification

We describe here a properly recent application of the Gröbner Cover algorithm (GC) providing an algebraic support to Dynamic Geometry computations of geometrical loci. It provides a complete algebraic solution of locus computation as well as a suitable taxonomy allowing to distinguish the nature of the different components. We included a new algorithm Locus into the Singular grobcov.lib library for this purpose. A web prototype has been implemented using it in Geogebra

Using Maple's RegularChains library to automatically classify plane geometric loci

We report a preliminary discussion on the usability of the RegularChains library of Maple for the automatic computation of plane geometric loci and envelopes in graphical interactive environments. We describe a simple implementation of a recently proposed taxonomy of algebraic loci, and its extension to envelopes of families of curves is also discussed. Furthermore, we sketch how currently unsolvable problems in interactive environments can be approached by using the RegularChains library

An algebraic taxonomy for locus computation in dynamic geometry

The automatic determination of geometric loci is an important issue in Dynamic Geometry. In Dynamic Geometry systems, it is often the case that locus determination is purely graphical, producing an output that is not robust enough and not reusable by the given software. Parts of the true locus may be missing, and extraneous objects can be appended to it as side products of the locus determination process. In this paper, we propose a new method for the computation, in dynamic geometry, of a locus defined by algebraic conditions. It provides an analytic, exact description of the sought locus, making possible a subsequent precise manipulation of this object by the system. Moreover, a complete taxonomy, cataloging the potentially different kinds of geometric objects arising from the locus computation procedure, is introduced, allowing to easily discriminate these objects as either extraneous or as pertaining to the sought locus. Our technique takes profit of the recently developed GröbnerCover algorithm. The taxonomy introduced can be generalized to higher dimensions, but we focus on 2-dimensional loci for classical reasons. The proposed method is illustrated through a web-based application prototype, showing that it has reached enough maturity as to be considered a practical option to be included in the next generation of dynamic geometry environments