Topology of 2D and 3D Rational Curves

In this paper we present algorithms for computing the topology of planar and space rational curves defined by a parametrization. The algorithms given here work directly with the parametrization of the curve, and do not require to compute or use the implicit equation of the curve (in the case of planar curves) or of any projection (in the case of space curves). Moreover, these algorithms have been implemented in Maple; the examples considered and the timings obtained show good performance skills.Comment: 26 pages, 19 figure

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 limit point and the T-function

Let P(t) ϵ P2 (K(t)) be a rational projective parametrization of a plane curve C. In this paper, we introduce the notion of limit point, PL, of P(t), and some remarkable properties of PL are obtained. In particular, if the singularities of C are P1, . . . , Pn and PL (all of them ordinary) and their respective multiplicities are m1, . . . , mn and mL, we show that T(s) = n i=1 HPi (s) m-1HPL (s) mL-1 , where T(s) is the univariate resultant of two polynomials obtained from P(t), and HP1 (s), . . . , HPn (s), HPL (s) are the fibre functions of the singularities. The fibre function of a point P is a polynomial HP (s) whose roots are the fibre of P. Thus, a complete classification of the singularities of a given plane curve, via the factorization of a resultant, is obtained.Ministerio de Ciencia, Innovación y Universidade

An in depth analysis, via resultants, of the singularities of a parametric curve

Let C be an algebraic space curve defined by a rational parametrization P(t)∈K(t)ℓ, ℓ≥2. In this paper, we consider the T-function, T(s), which is a polynomial constructed from P(t) by means of a univariate resultant, and we show that T(s) contains essential information concerning the singularities of C. More precisely, we prove that T(s)=∏i=1nHPi(s), where Pi, i=1,…,n, are the (ordinary and non-ordinary) singularities of C and HPi, i=1,…,n, are polynomials, each of them associated to a singularity, whose factors are the fiber functions of those singularities as well as those other belonging to their corresponding neighborhoods. That is, HQ(s)=HQ(s)m−1∏j=1kHQj(s)mj−1, where Q is an m-fold point, Qj,j=1,…,k, are the neighboring singularities of Q, and mj,j=1,…,k, are their corresponding multiplicities (HP denotes the fiber function of P). Thus, by just analyzing the factorization of T, we can obtain all the singularities (ordinary and non-ordinary) as well as interesting data relative to each of them, like its multiplicity, character, fiber or number of associated tangents. Furthermore, in the case of non-ordinary singularities, we can easily get the corresponding number of local branches and delta invariant.Ministerio de Ciencia, Innovación y Universidade