Covering of surfaces parametrized without projective base points

Abstract

This is the author's version of the work. It is posted here for your personal use. Not for redistribution. The definitive Version of Record was published in "Sendra J.R., Sevilla D., Villarino C. Covering of surfaces parametrized without projective base points. Proc. ISSAC2014 ACM Press, pages 375-380, 2014,\ud ISBN:978-1-4503-2501-1". http://dx.doi.org/10.1145/2608628.2608635We prove that every a ne rational surface, parametrized by means of an a ne rational parametrization without projective base points, can be covered by at most three parametrizations.\ud Moreover, we give explicit formulas for computing the coverings. We provide two di erent approaches: either\ud covering the surface with a surface parametrization plus a curve parametrization plus a point, or with the original parametrization plus two surface reparametrizations of it

Covering rational ruled surfaces

“…in the surface case, with four pieces. In [SSV14a] we show that, if a surface admits a rational parametrization without projective base points, then it can be covered with at most three pieces. Continuing with this research, in this paper we analyze the problem of covering rational ruled surfaces.…”

Covering Rational Surfaces with Rational Parametrization Images

“…Rational parametrizations of algebraic varieties are an important tool in many geometric applications like those in computer aided design (see, e.g., [1,2]) or computer vision (see, e.g., [3,4]). Nevertheless, the applicability of this tool can be negatively affected if the parametrization is missing basic properties: for instance its injectivity, its surjectivity, or the nature of the ground field where the coefficients belong to; see, e.g., the introductions of the papers [5][6][7] for some illustrating examples of this phenomenon.…”

“…Let us mention here some illustrating situations of this phenomenon; for more detailed examples, we refer the reader, e.g., to the introductions of the papers [5][6][7]. Let us say that we are given a rational parametrization P (t) of a curve that describes the possible positions that a given robot may achieve.…”

“…In [7] it is proved that ruled surfaces can be covered by using two rational parametrizations. In addition, in [6] an algorithm to cover an affine rational surface without based points at infinity with at most three parametrizations is presented.…”

On the existence of birational surjective parametrizations of affine surfaces

“…, f s of X such that the union of their imagines does cover the whole affine surface, that is ∪ s i=1 f i (C 2 ) = X (see e.g. [3], [7], [18], [21], [20]). Nevertheless, the following natural question arises: does there exist a surjective birational affine parametrization for every rational affine surface?…”