An algebraic taxonomy for locus computation in dynamic geometry

Abstract

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

Computing envelopes in dynamic geometry environments

“…Thus, the standard DG approach to find the geometric locus of a (tracer) point depending somehow on another (mover) point consists of automatically sampling the path of the mover, and, for each sample, compute the actual position of the corresponding tracer point. The list of computed points is then plotted, showing the sought trajectory (see [6] for a full description of loci finding strategies in DG). Replacing the tracer point by a tracing curve, Cabri and Cinderella commands are then used to plot a collection of curves in order to suggest the envelope.…”

“…Finally, the last part of Sections 3 and 4 address the second objective of this note, namely, to describe the basic issues behind a new command for envelope computation, featured in the new 5.0 version of GeoGebra (September 2014) and based on a series of recent contributions by the authors of this note and their collaborators [5][6][7][8][9][10][11]. The idea here is to present just a sketchy picture on how some key ingredients from effective algebraic geometry are put together to conform the algorithmic approach behind this GeoGebra command.…”

Software Using the Gröbner Cover for Geometrical Loci Computation and Classification

“…Montes implemented in Singular the grobcov.lib library [7], whose actual version incorporates Kapur-Sun-Wang algorithm [2] for computing the initial Gröbner System used in grobcov algorithm, as described in [4], and recently also the Locus algorithm described here. A more detailed description can be seen in [1].…”

“…where Si = ∪jCij and Cij = {(pij, {p ijk : The taxonomy that we give is motivated by the interpretation of the solutions in a lot of loci problems of different kind (see [1]):…”

Using Maple’s RegularChains Library to Automatically Classify Plane Geometric Loci

“…Nevertheless, since Zariski closures are obtained as result, loci equations can include spurious points, and complete 1-dimensional objects can even appear due to construction degeneracies. A finer analysis of loci problems can be done through a recently proposed taxonomy [2], and it has been implemented using the GröbnerCover algorithm [3]. Here we describe an alternative implementation of the taxonomy using Maple's RegularChains library.…”