On the problem of proper reparametrization for rational curves and surfaces

A rational parametrization of an algebraic curve (resp. surface) establishes a rational correspondence of this curve (resp. surface) with the affine or projective line (resp. affine or projective plane). This correspondence is a birational equivalence if the parametrization is proper. So, intuitively speaking, a rational proper parametrization trace the curve or surface once. We consider the problem of computing a proper rational parametrization from a given improper one. For the case of curves we generalize, improve and reinterpret some previous results. For surfaces, we solve the problem for some special surface's parametrizations

Properness and inversion of rational parametrizations of surfaces

In this paper we characterize the properness of rational parametrizations of hypersurfaces by means of the existence of intersection points of some additional algebraic hypersurfaces directly generated from the parametrization over a field of rational functions. More precisely, if V is a hypersurface over an algebraically closed field ? of characteristic zero and is a rational parametrization of V, then the characterization is given in terms of the intersection points of the hypersurfaces defined by x i q i (t¯)−p i (t¯), i=1,...,n over the algebraic closure of ?(V). In addition, for the case of surfaces we show how these results can be stated algorithmically. As a consequence we present an algorithmic criteria to decide whether a given rational parametrization is proper. Furthermore, if the parametrization is proper, the algorithm also computes the inverse of the parametrization. Moreover, for surfaces the auxiliary hypersurfaces turn to be plane curves over ?(V), and hence the algorithm is essentially based on resultants. We have implemented these ideas, and we have empirically compared our method with the method based on Gröbner basis

A survey of the representations of rational ruled surfaces

The rational ruled surface is a typical modeling surface in computer aided geometric design. A rational ruled surface may have different representations with respective advantages and disadvantages. In this paper, the authors revisit the representations of ruled surfaces including the parametric form, algebraic form, homogenous form and Pl¨ucker form. Moreover, the transformations between these representations are proposed such as parametrization for an algebraic form, implicitization for a parametric form, proper reparametrization of an improper one and standardized reparametrization for a general parametrization. Based on these transformation algorithms, one can give a complete interchange graph for the different representations of a rational ruled surface. For rational surfaces given in algebraic form or parametric form not in the standard form of ruled surfaces, the characterization methods are recalled to identify the ruled surfaces from them.Agencia Estatal de Investigació

Cissoid constructions of augmented rational ruled surfaces

J.R. Sendra is member of the Research Group ASYNACS (Ref.CT-CE2019/683)Given two real affine rational surfaces we derive a criterion for deciding the rationality of their cissoid. Furthermore, when one of the surfaces is augmented ruled and the other is either an augmented ruled or an augmented Steiner surface, we prove that the cissoid is rational. Furthermore, given rational parametrizations of the surfaces, we provide a rational parametrization of the cissoid.Ministerio de Economía y CompetitividadEuropean Regional Development Fun

A Lie Algebra Method for Rational Parametrization of Severi-Brauer Surfaces

It is well-known that a Severi-Brauer surface has a rational point if and only if it is isomorphic to the projective plane. Given a Severi-Brauer surface, we study the problem to decide whether such an isomorphism to the projective plane, or such a rational point, does exist; and to construct such an isomorphism or such a point in the affirmative case. We give an algorithm using Lie algebra techniques. The algorithm has been implemented in Magma.Comment: 16 pages some minor revision

Parametrization of aproximate algebraic surfaces by lines

In this paper we present an algorithm for parametrizing approximate algebraic surfaces by lines. The algorithm is applicable to ²-irreducible algebraic surfaces of degree d having an ²–singularity of multiplicity d−1, and therefore it generalizes the existing approximate parametrization algorithms. In particular, given a tolerance ² > 0 and an ²-irreducible algebraic surface V of degree d, the algorithm computes a new algebraic surface V , that is rational, as well as a rational parametrization of V . In addition, in the error analysis we show that the output surface V and the input surface V are close. More precisely, we prove that V lies in the offset region of V at distance, at most, O(² 1 2d )