A partial solution to the problem of proper reparametrization for rational surfaces☆,☆☆
Introduction
In this paper, we deal with the reparametrization problem, that is, with the problem of computing a rational proper reparametrization of a given improperly parametrized algebraic surface. More precisely, given an algebraically closed field , and , , a rational parametrization of a surface , the reparametrization problem consists in computing a proper parametrization of , , and , such that .
Although it is known from Castelnuovoʼs Theorem that unirationality and rationality are equivalent over algebraically closed fields, only partial results approaching the problem algorithmically are known (see Pérez-Díaz, 2006). In particular, given an algebraically closed field , and a rational parametrization of a surface , an algorithm is presented in Pérez-Díaz (2006) to determine whether there exists such that , and is a proper parametrization of . In the affirmative case, R and are computed.
The approach presented in this paper complements the results obtained in Pérez-Díaz (2006). More precisely, in Pérez-Díaz (2006), the reparametrization problem is solved for those surfaces parametrized by that admit R of the form , and such that . In this paper, we deal with surfaces not necessarily satisfying this condition. In addition, for those surfaces for which a rational proper reparametrization is not found, we show how to decrease the degree of the rational map induced by the parametrization. For this purpose, we need that at least one of two auxiliary parametrizations defined from is not proper.
The reparametrization problem, in particular when the variety is a curve or a surface, is especially interesting in some practical applications in Computer Aided Geometric Design (CAGD) where objects are often given and manipulated parametrically. In addition, proper parametrizations play an important role in many practical applications in CAGD, such as in visualization (see Hoffmann et al., 1997 or Hoschek and Lasser, 1993) or rational parametrization of offsets (see Arrondo et al., 1997). Also, it is provided an implicitization approach based on resultants (see Cox et al., 1998, and Sendra and Winkler, 2001).
A direct approach to the reparametrization problem could consist in first implicitizing the parametrization (see Busé et al., 2003, Cox, 2001, Kotsireas, 2004, Sendra and Winkler, 2001), and then to apply algorithms developed for instance in Cox et al. (1997), Goldman et al. (1984), González-Vega (1997), van Hoeij (1997), Hoffmann et al. (1997), Schicho (1998), Sendra and Winkler, 1991, Sendra and Winkler, 1997, to parametrize the implicit equation. However, some of these implicitization methods have difficulties in the presence of base points, or deal only with special cases or, although always valid, the computing time is not totally satisfactory. In Pérez-Díaz and Sendra (2008), an algorithm is presented, based on polynomial gcds and univariate resultants, that is always valid. However, even with this approach, the solution is, in most of cases, too time consuming (see Section 3.1).
Therefore, we would like to approach the problem by means of rational reparametrizations. By rational reparametrization we basically mean without implicitizing, or more formally, by finding a nonconstant rational change of parameter, if it exists, that transforms the input parametrization into a new parametrization of the same curve or surface. Note that any reparametrization of a rational parametrization is again a parametrization of the same variety.
It is well known that for the case of curves, it is always possible to reparametrize an improperly parametrized curve in such a way that it becomes properly parametrized. In Alonso et al. (1995), Gutierrez et al. (2002), Pérez-Díaz (2006) and Sederberg (1986), some approaches are presented to compute a proper parametrization from a given improper one.
The approach presented in this paper deals with the surface case, and it is based on polynomial gcds and univariate resultants. The computing time is very satisfactory (see Section 3.1). More precisely, the algorithm presented follows from the algorithm Proper Reparametrization for Space Curves developed in Section 2 and derived from the results in Pérez-Díaz (2006). The basic idea of the approach presented in this paper is to compute a reparametrization of two auxiliary parametrizations of two space curves, , , obtained directly from a given rational parametrization of the surface defined over an algebraically closed field (see Definition 1). Moreover, since when we compose two rational maps we multiply their degrees, we can deduce some properties that relate the degree of the rational map induced by the given parametrization to the degree of the output parametrization , and the degree of the rational maps induced by the two auxiliary parametrizations, and . Furthermore, we also show the relation of the degrees of the rational maps induced by and with the degree of with respect to the variables , .
The structure of the paper is as follows: In Section 2, we present an algorithm for computing a proper reparametrization of an algebraic space curve. The algorithm is derived from the results presented in Pérez-Díaz (2006). In Section 3, we outline the algorithm that solves (in some cases) the problem of computing a rational proper reparametrization for a given improperly parametrized algebraic surface. More precisely, we introduce some auxiliary partial parametrizations defined from the input rational parametrization (see Definition 1), and we prove a theorem (see Theorem 3), where we characterize the properness of in terms of the properness of its partial parametrizations. The idea provided by this theorem will be used to derive the algorithm and to characterize the properness of the output reparametrization (see Theorem 4 and Corollary 3). In addition, for those surfaces for which we cannot find a rational proper reparametrization, if at least one of two auxiliary parametrizations defined from is not proper, we show how to compute a rational reparametrization such that the degree of the rational map induced by it is less than the degree induced by the input parametrization (see Corollary 2). Finally, we present the actual computing times of the implementation, and we show that the algorithm presented here is much more efficient and powerful, than first finding the implicit equation (see Section 3.1). Section 4 is devoted to summarizing the contributions of the paper, and we comment on how the results presented in the paper can easily be extended to a variety of dimension 2, rationally parametrized by where is an algebraically closed field.
Section snippets
Proper reparametrization for space curves
The problem of proper reparametrization for curves can be stated as follows: given a field K (not necessarily an algebraically closed field), and a rational parametrization of an algebraic curve , find a rational proper parametrization of , and a rational function such that .
A parametrization of is proper if and only if the map is birational, or equivalently, if for almost every point on and for almost all values of the
Proper reparametrization for surfaces
In Section 2, we dealt with the problem of computing a rational proper reparametrization of a given improperly parametrized algebraic space curve. For the case of surfaces, although it is known from Castelnuovoʼs Theorem that unirationality and rationality are equivalent over algebraically closed fields, the problem is not solved computationally. That is, there does not exist an algorithm that computes the proper reparametrization.
In this section, given an algebraically closed field , and
Conclusion
In this paper, we study the reparametrization problem. That is, given an algebraically closed field , and a rational parametrization of an algebraic surface , we consider the problem of computing a proper rational parametrization from .
The approach presented complements some previous results developed in Pérez-Díaz (2006). More precisely, in Pérez-Díaz (2006), the reparametrization problem is solved for those surfaces parametrized by such that there exists
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This paper has been recommended for acceptance by Hans Hagen.
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Member of the Research Group asynacs (Ref. ccee2011/r34).