Sonia Pérez-Díaz

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 surfaces parametrizations.

Parametrization of approximate algebraic curves by lines

It is well known that irreducible algebraic plane curves having a singularity of maximum multiplicity are rational and can be parametrized by lines. In this paper, given a tolerance e > 0 and an e-irreducible algebraic plane curve L of degree d having an e-singularity of multiplicity d - 1, we provide an algorithm that computes a proper parametrization of a rational curve that is exactly parametrizable by lines. Furthermore, the error analysis shows that under certain initial conditions that ensures that points are projectively well defined, the output curve lies within the offset region of L at distance at most 2√2e1(2d)exp(2).

Properness and Inversion of Rational Parametrizations of Surfaces

Abstract. 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 xiqi(t¯)−pi(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.

Approximate parametrization of plane algebraic curves by linear systems of curves

It is well known that an irreducible algebraic curve is rational (i.e. parametric) if and only if its genus is zero. In this paper, given a tolerance @e>0 and an @e-irreducible algebraic affine plane curve C of proper degree d, we introduce the notion of @e-rationality, and we provide an algorithm to parametrize approximately affine @e-rational plane curves by means of linear systems of (d-2)-degree curves. The algorithm outputs a rational parametrization of a rational curve C@? of degree d which has the same points at infinity as C. Moreover, although we do not provide a theoretical analysis, our empirical analysis shows that C@? and C are close in practice.

Computation of the singularities of parametric plane curves

Given an algebraic plane curve C defined by a rational parametrization P(t), we present formulae for the computation of the degree of C, and the multiplicity of a point. Using the results presented in [Sendra, J.R., Winkler, F., 2001. Tracing index of rational curve parametrizations. Computer Aided Geometric Design 18 (8), 771-795], the formulae simply involve the computation of the degree of a rational function directly determined from P(t). Furthermore, we provide a method for computing the singularities of C and analyzing the non-ordinary ones without knowing its defining polynomial. This approach generalizes the results in [Abhyankar, S., 1990. Algebraic geometry for scientists and engineers. In: Mathematical Surveys and Monographs, vol. 35. American Mathematical Society; van den Essen, A., Yu, J.-T., 1997. The D-resultants, singularities and the degree of unfaithfulness. Proceedings of the American Mathematical Society 25, 689-695; Gutierrez, J., Rubio, R., Yu, J.-T., 2002. D-Resultant for rational functions. Proceedings of the American Mathematical Society 130 (8), 2237-2246] and [Park, H., 2002. Effective computation of singularities of parametric affine curves. Journal of Pure and Applied Algebra 173, 49-58].

Parametrization of approximate algebraic surfaces by lines

In this paper we present an algorithm for parametrizing approximate algebraic surfaces by lines. The algorithm is applicable to e-irreducible algebraic surfaces of degree d having an e-singularity of multiplicity d - 1, and therefore it generalizes the existing approximate parametrization algorithms. In particular, given a tolerance e > 0 and an e-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(e1/(2d)).

Characterization of rational ruled surfaces

The algebraic ruled surface is a typical modeling surface in computer aided geometric design. In this paper, we present algorithms to determine whether a given implicit or parametric algebraic surface is a rational ruled surface, and in the affirmative case, to compute a standard parametric representation for the surface.

Distance bounds of ε-points on hypersurfaces

e-Points were introduced by the authors (see [S. Perez-Diaz, J.R. Sendra, J. Sendra, Parametrization of approximate algebraic curves by lines, Theoret. Comput. Sci. 315(2-3) (2004) 627-650 (Special issue); S. Perez-Diaz, J.R. Sendra, J. Sendra, Parametrization of approximate algebraic surfaces by lines, Comput. Aided Geom. Design 22(2) (2005) 147-181; S. Perez-Diaz, J.R. Sendra, J. Sendra, Distance properties of e-points on algebraic curves, in: Series Mathematics and Visualization, Computational Methods for Algebraic Spline Surfaces, Springer, Berlin, 2005, pp. 45-61]) as a generalization of the notion of approximate root of a univariate polynomial. The notion of e-point of an algebraic hypersurface is quite intuitive. It essentially consists in a point such that when substituted in the implicit equation of the hypersurface gives values of small module. Intuition says that an e-point of a hypersurface is a point close to it. In this paper, we formally analyze this assertion giving bounds of the distance of the e-point to the hypersurface. For this purpose, we introduce the notions of height, depth and weight of an e-point. The height and the depth control when the distance bounds are valid, while the weight is involved in the bounds.

A FIRST APPROACH TOWARDS NORMAL PARAMETRIZATIONS OF ALGEBRAIC SURFACES

In this paper we analyze the problem of deciding the normality (i.e. the surjectivity) of a rational parametrization of a surface . The problem can be approached by means of elimination theory techniques, providing a proper close subset where surjectivity needs to be analyzed. In general, these direct approaches are unfeasible because is very complicated and its elements computationally hard to manipulate. Motivated by this fact, we study ad hoc computational alternative methods that simplifies . For this goal, we introduce the notion of pseudo-normality, a concept that provides necessary conditions for a parametrization for being normal. Also, we provide an algorithm for deciding the pseudo-normality. Finally, we state necessary and sufficient conditions on a pseudo-normal parametrization to be normal. As a consequence, certain types of parametrizations are shown to be always normal. For instance, pseudo-normal polynomial parametrizations are normal. Moreover, for certain class of parametrizations, we derive an algorithm for deciding the normality.

Partial degree formulae for rational algebraic surfaces

In this paper, we present formulae for the computation of the partial degrees w.r.t. each variable of the implicit equation of a rational surface given by means of a proper parametrization. Moreover, when the parametrization is not proper we give upper bounds. These formulae generalize the results in [17] to the surface case, and they are based on the computation of the degree of the rational maps induced by the projections, onto the coordinate planes of the three dimensional space, of the input surface parametrization. In addition, using the results presented in [9] and [10], the formulae simply involve the computation of the degree of univariate polynomials directed determined from the parametrization by means of some univariate resultants and some polynomial gcds.