A critical set of revolution surface parametrizations

Given the classical rational arametrization of a surface of revolution, generated by rotating a rational curve around the z-axis, we determine a superset containing all the possible points of the surface non-reachable by the parametrization; that is a critical set of the parametrization

A critical set of revolution surface parametrizations

Given the classical rational arametrization of a surface of revolution, generated by rotating a rational curve around the z-axis, we determine a superset containing all the possible points of the surface non-reachable by the parametrization; that is a critical set of the parametrization

Covering of surfaces parametrized without projective base points

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

On the existence of birational surjective parametrizations of affine surfaces

In this paper we show that not all affine rational complex surfaces can be parametrized birational and surjectively. For this purpose, we prove that, if S is an affine complex surface whose projective closure is smooth, a necessary condition for S to admit a birational surjective parametrization from an open subset of the affine complex plane is that the curve at infinity of S must contain at least one rational component. As a consequence of this result we provide examples of affine rational surfaces that do not admit birational surjective parametrizations.Ministerio de Economía y CompetitividadJunta de ExtremaduraEuropean Regional Development Fun

Sobre corbes paramètriques i polígons de Newton

Les corbes i superfícies algebraiques poden ser definides implícitament com a solucions d'equacions polinomials i, de vegades, també poden definir-se paramètricament, mitjançant funcions racionals. Plantegem el problema de la conversió d'una d'aquestes formes de representació a l'altra. A continuació, explorem la possibilitat d'obtenir, a partir de les equacions paramètriques i sense necessitat d'efectuar l'operació costosa de la implicitació, un objecte pròxim a les equacions implícites associades: el polítop de Newton d'una hipersuperfície donada paramètricament.Algebraic curves and surfaces can be defined as solutions of polynomial equations and, sometimes, by parametric equations of rational functions as well. We consider the problem of moving from parametric to implicit representations, a usually involved process. We also explore the possibility of obtaining an object close to the implicit equations from the parametric ones: the Newton polytop of a hypersurface given in parametric form

Covering rational ruled surfaces

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.Ministerio de Economía y Competitivida

Sufficient conditions for the surjectivity of radical curve parametrizations

J. Caravantes and C. Villarino belong to the Research Group ASYNACS (Ref. CT-CE2019/683). D. Sevilla is a member of the research group GADAC and is partially supported by Junta de Extremadura and Fondo Europeo de Desarrollo Regional (GR21055).En este artículo se introduce la noción de parametrización radical suprayectiva y se dan condiciones suficientes para que una parametrización radical de una curva sea suprayectiva.In this paper, we introduce the notion of surjective radical parametrization and we prove sufficient conditions for a radical curve parametrization to be surjective.Agencia Estatal de Investigació