Computing Locus Equations for Standard Dynamic Geometry Environments

Abstract

Abstract. GLI (Geometric Locus Identifier), an open web-based tool to determine equations of geometric loci specified using Cabri Geometry and The Geometer's Sketchpad, is described. A geometric construction of a locus is uploaded to a Java Servlet server, where two computer algebra systems, CoCoA and Mathematica, following the Groebner basis method, compute the locus equation and its graph. Moreover, an OpenMath description of the geometric construction is given. GLI can be efficiently used in mathematics education, as a supplement of the locus functions of the standard dynamic geometry systems. The system is located at

Some issues on the automatic computation of plane envelopes in interactive environments

“…Bearing this remark in mind, let us discuss here an absurd, toy example, about such situation. For this purpose let us consider, again, a family of lines in the plane, each one going through a point of the curve t 3 1 + t 2 2 = 1 and so that they are all parallel to the x-axis. Each point (t 1 , t 2 ) parametrizes the family of lines y = t 2 .…”

“…Each point (t 1 , t 2 ) parametrizes the family of lines y = t 2 . Geometrically, the fact that t 2 is the second coordinate of an arbitrary point in the curve does not play any special role, since the family of lines is identical to that previously introduced, described as y = t for a single parameter t. Thus, it could happen that a naive dynamic geometry user could end up considering that computing the envelope for the constructed family y = t 2 , where t 3 1 +t 2 2 −1 = 0, will output the same envelope as for the family y = t, for a single parameter t.…”

“…That is, the points in this plane that lift up to points of V such that either they are singular or they have tangent plane parallel to the (t 1 , t 2 )-plane. Namely, we should consider the projection of the set of points in V where the Jacobian of y = t 2 , t 3 1 + t 2 2 − 1 = 0, with respect to (t 1 , t 2 ), vanishes, i.e. where 3t 2 1 = 0.…”

On the Parametric Representation of Dynamic Geometry Constructions

“…For loci whose points are of very different magnitudes, numerical errors appear very rapidly as the degree increases" [21]. Although this algorithm has not been made public by the vendor, it seems that it is very unstable, and the returned results are frequently erroneous even for simple cases [22]. For instance, computing an astroid as the envelope of a moving segment with each end on one of a pair of perpendicular axes, Cabri gives different equations when constraining the segment to the upper and lower half planes, as shown in Fig.…”

A Maple Package for Automatic Theorem Proving and Discovery in 3D-Geometry

“…-a connection between the DGS Cabry Geometry and GSP and the CAS Mathematica and CoCoA for geometric loci equations finding, denoted GLI [28]. It uses the standard OpenMath [29] for the communication between the DGS and the CAS -a connection between the new version of the 3D-DGS Calques3D [30] and the CAS CoCoA and Mathematica, denoted 3D-LD [31].…”