We present an algorithm that covers any given rational ruled surface with two rational parametrizations. In addition, we present an algorithm that transforms any rational surface parametrization into a new rational surface parametrization without affine base points and such that the degree of the corresponding maps is preserved.
In this article algebraic constructions are introduced in order to study the
variety defined by a radical parametrization (a tuple of functions involving
complex numbers, $n$ variables, the four field operations and radical
extractions). We provide algorithms to implicitize radical parametrizations and
to check whether a radical parametrization can be reparametrized into a
rational parametrization.Comment: 26 pages; revised version accepted for publication in Computer Aided
Geometric Desig
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
Abstract. This paper deals with a family of spatial rational curves that were introduced in 1999 by Andradas, Recio, and Sendra, under the name of hypercircles, as an algorithmic cornerstone tool in the context of improving the rational parametrization (simplifying the coefficients of the rational functions, when possible) of algebraic varieties. A real circle can be defined as the image of the real axis under a Moebius transformation in the complex field. Likewise, and roughly speaking, a hypercircle can be defined as the image of a line ("the K-axis") in an n-degree finite algebraic extension K(α) ≈ K n under the transformationThe aim of this article is to extend, to the case of hypercircles, some of the specific properties of circles. We show that hypercircles are precisely, via K-projective transformations, the rational normal curve of a suitable degree. We also obtain a complete description of the points at infinity of these curves (generalizing the cyclic structure at infinity of circles). We characterize hypercircles as those curves of degree equal to the dimension of the ambient affine space and with infinitely many K-rational points, passing through these points at infinity. Moreover, we give explicit formulae for the parametrization and implicitation of hypercircles. Besides the intrinsic interest of this very special family of curves, the understanding of its properties has a direct application to the simplification of parametrizations problem, as shown in the last section.
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