Title: A symbolic-numeric environment for computing equidistant curves

Authors: Francisco Botana
           (Univ. de Vigo, Spain)

           Miguel Á. Abánades
           (Univ. Complutense de Madrid, Spain)

Extended Abstract:

The bisector of two geometrical elements (points, curves, surfaces, etc.)
is the locus set of points equidistant to them. While the
concept of a bisector is simple, its effective computation is usually
not trivial, involving a non algebraic approach in an essential way.
In this talk we will show a web-based system that completely
determines bisectors of points and algebraic curves (provided as graph
of functions, by implicit functions, or as the conic through five
points, for instance).

The algorithms behind the bisector computations use on one hand
Groebner-based elimination to determine an algebraic variety
containing the bisector and on the other hand a numeric approach to
provide the graph of the semi-algebraic subset corresponding to the
true bisector. The system mainly consists of an interactive drawing
canvas where the algebraic and true bisectors are displayed together
with the point and curve determined by the user. Moreover, the
equation of the algebraic bisector is provided in text form.

An important feature of the system is that it is solely based on open
source software. It uses the dynamic geometry system GeoGebra and the
computer algebra system Sage. More concretely, the algebraic knowledge
introduced through GeoGebra is remotely processed by Sage and its
Singular component by means of its efficient elimination algorithms.
Moreover, the recent GroebnerCover algorithm is used for the exact
determination of segment endpoints in the case of true bisectors.

Partially supported by grants MTM2008-04699-C03-03/MTM and
MTM2011-25816-C02-02  from the Spanish MINECO.