Analysis and construction of rational curve parametrizations with non-ordinary singularities

In this paper, we provide a method that allows to construct parametric curves having (or not) non-ordinary singularities and having (or not) neighboring points. This method is based on a characterization of the non-ordinary singularities and neighboring points by means of linear equations involving the given parametrization. As a consequence, we obtain an algorithm that constructs a parametrization which contains a given point, P, as a singularity as well as some additional information as for instance, the order of P, parameters corresponding to P, multiplicity of each parameter and the singularities in the first neighborhood of the singularity P.Ministerio de Economía y Competitivida

The T-function of a parametric curve

In this paper, we introduce the T–function, T(s), which is a polynomial defined by means of a univariate resultant constructed from a given parametrization P(t) ∈ K(t) n , n ≥ 2 of an algebraic space curve C. It is shown that T(s) = Qn i=1 HPi (s) mi−1 , where HPi (s), i = 1, . . . , n are polynomials (the fibre functions) whose roots are the fibre of the ordinary singularities Pi ∈ C of multiplicity mi , i = 1, . . . , n of C. Therefore, a complete classification of the singularities of C, via the factorization of a resultant, is obtained.Agencia Estatal de Investigació

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

On the computation of singularities of parametrized ruled surfaces

Given a ruled surface V defined in the standard parametric form P(t1, t2), we present an algorithm that determines the singularities (and their multiplicities) of V from the parametrization P. More precisely, from P we construct an auxiliary parametric curve and we show how the problem can be simplified to determine the singularities of this auxiliary curve. Only one univariate resultant has to be computed and no elimination theory techniques are necessary. These results improve some previous algorithms for detecting singularities for the special case of parametric ruled surfaces.Ministerio de Ciencia, Innovación y Universidade

A partial solution to the problem of proper reparametrization for rational surfaces

Given an algebraically closed field K, and a rational parametrization P of an algebraic surface V ⊂ K3 , we consider the problem of computing a proper rational parametrization Q from P (reparametrization problem). More precisely, we present an algorithm that computes a rational parametrization Q of V such that the degree of the rational map induced by Q is less than the degree induced by P. The properness of the output parametrization Q is analyzed. In particular, if the degree of the map induced by Q is one, then Q is proper and the reparametrization problem is solved. The algorithm works if at least one of two auxiliary parametrizations defined from P is not proper

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].Ministerio de Educación y CienciaComunidad de MadridUniversidad de Alcal