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 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.
Highlights
A geometric locus is a set of points satisfying some condition
In Dynamic Geometry (DG), the term locus generally refers to loci of this second kind: i.e. to the trajectory determined by the different positions of a 10 point, corresponding to the different instances of the construction determined by the different positions of a second point along the path to which where it is constrained
This is the case for the first standard DG systems developed in the late 80’s, but it is true for 15 more recent ones, such as GeoGebra [3] or Java Geometry Expert [4]
Summary
A geometric locus is a set of points satisfying some condition. For instance, the set of points A at a given distance d to a specific point C is the circle centered at C of radius d. In Dynamic Geometry (DG), the term locus generally refers to loci of this second kind: i.e. to the trajectory determined by the different positions of a 10 point (the tracer, as point P above), corresponding to the different instances of the construction determined by the different positions of a second point (the mover, such as point Q above) along the path to which where it is constrained. The last property is achieved by developing an elaborated taxonomy for different pieces of the aforementioned projection set, and by algorithmically assigning to each one the corresponding label (see Section 3) Following this taxonomy, we establish a protocol that yields a faithful sym bolic description of a given locus in terms of constructible sets, collecting pieces of the projection set featuring ‘good’ labels. The provided examples show that our method overcomes limitations found 55 in previous proposals, and that it allows the computation of generalized loci in the sense of [6], see Section 5.4
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