On the problem of proper reparametrization for rational curves and surfaces (original) (raw)

A First Approach Towards Normal Parametrizations of Algebraic Surfaces

International Journal of Algebra and Computation, 2010

In this paper we analyze the problem of deciding the normality (i.e. the surjectivity) of a rational parametrization of a surface [Formula: see text]. The problem can be approached by means of elimination theory techniques, providing a proper close subset [Formula: see text] where surjectivity needs to be analyzed. In general, these direct approaches are unfeasible because [Formula: see text] is very complicated and its elements computationally hard to manipulate. Motivated by this fact, we study ad hoc computational alternative methods that simplifies [Formula: see text]. 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-...

Points on algebraic curves and the parametrization problem

Lecture Notes in Computer Science, 1997

A plane algebraic curve is given as the zeros of a bivariate polynomial. However, this implicit representation is badly suited for many applications, for instance in computer aided geometric design. What we want in many of these applications is a rational parametrization of an algebraic curve. There are several approaches to deciding whether an algebraic curve is parametrizable and if so computing a parametrization. In all these approaches we ultimately need some simple points on the curve. The eld in which we can nd such points crucially in uences the coe cients in the resulting parametrization. We show how to nd such simple points over some practically interesting elds. Consequently, we are able to decide whether an algebraic curve de ned over the rational numbers can be parametrized over the rationals or the reals. Some of these ideas also apply to parametrization of surfaces. If in the term geometric reasoning we do not only include the process of proving or disproving geometric statements, but also the analysis and manipulation of geometric objects, then algorithms for parametrization play an important role in this wider view of geometric reasoning.

Computation of the degree of rational surface parametrizations

Journal of Pure and Applied Algebra, 2004

A rational a ne parametrization of an algebraic surface establishes a rational correspondence of the a ne plane with the surface. We consider the problem of computing the degree of such a rational map. In general, determining the degree of a rational map can be achieved by means of elimination theoretic methods. For curves, it is shown that the degree can be computed by gcd computations. In this paper, we show that the degree of a rational map induced by a surface parametrization can be computed by means of gcd and univariate resultant computations. The basic idea is to express the elements of a generic ÿbre as the ÿnitely many intersection points of certain curves directly constructed from the parametrization, and deÿned over the algebraic closure of a ÿeld of rational functions.

On parameterizations of plane rational curves and their syzygies

Mathematische Nachrichten, 2015

Let C be a plane rational curve of degree d and p :C → C its normalization. We are interested in the splitting type (a, b) of C, where O P 1 (−a − d) ⊕ O P 1 (−b − d) gives the syzigies of the ideal (f 0 , f 1 , f 2) ⊂ K[s, t], and (f 0 , f 1 , f 2) is a parameterization of C. We want to describe in which cases (a, b) = (k, d − k) (2k ≤ d), via a geometric description; namely we show that (a, b) = (k, d − k) if and only if C is the projection of a rational curve on a rational normal surface in P k+1 .

Automatic parameterization of rational curves and surfaces IV: algebraic space curves

ACM Transactions on Graphics, 1989

For an irreducible algebraic space curve C that is implicitly defined as the intersection of two algebraic surfaces, f (x, y, z) = 0 and g(r, y, z) = 0, there always exists a birational correspondence between the points of C and the points of an irreducible plane curve P, whose genus is the same as that of C. Thus C is rational iff the genus of P is zero. Given an irreducible space curve C = ( f n g), with f and g not tangent along C, we present a method of obtaining a projected irreducible plane curve P together with birational maps between the points of P and C. Together with [4], this method yields an algorithm to compute the genus of C, and if the genus is zero, the rational parametric equations for C. As a biproduct, this method also yields the implicit and parametric equations of a rational surface S containing the space curve C.

On the existence of birational surjective parametrizations of affine surfaces

Journal of Algebra, 2018

In this paper we show that not all affine rational complex surfaces can be parametrized birational and surjectively. For this purpose, we prove that, if S is an affine complex surface whose projective closure is smooth, a necessary condition for S to admit a birational surjective parametrization from an open subset of the affine complex plane is that the infinity curve of S must contain at least one rational component. As a consequence of this result we provide examples of affine rational surfaces that do not admit birational surjective parametrizations.

Parametrization of approximate algebraic surfaces by lines

Computer Aided Geometric Design, 2005

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 ¿ 0 and an -irreducible algebraic plane curve C of degree d having an -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 deÿned, the output curve lies within the o set region of C at distance at most 2 √ 2 1=(2d) exp(2).

A relatively optimal rational space curve reparametrization algorithm through canonical divisors

Proceedings of the 1997 international symposium on Symbolic and algebraic computation - ISSAC '97, 1997

Let K be a given computable field of characteristic zero and let ILbe a finite field extension of K, with algebraic closure F. Assume a rational parametrization P(t) E L(t)nof some irreducible curve C in the affine n-space over F is also given. In this paper we will show, first, how to decide-without imp]icitization algorithms-whether the given curve C is definable (by a set of equations with coefficients) over K; and, if this is the case, we will determine-without computing the implicit, equation set and then using parametrization techniques a reparametrizatiou of P(t) over the smallest possible field extension of K; that is, over a field extension of K of degree at most two. 1 third author supported by HCM''SAC" and DGICYT PB 95.0563: "Sistenlas de Eruaciones Algebraicas: Resoluci6n y Aplicaciones" aud UAWPmy F~oIo/97: 'rAlgoritmos y aplicac iones de Ia.s variedades parar116tricas en diseiio geom&rico" I'm-mission to nlake digitcd/hard copy of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage, the copyright notice, the title of the publication and its date appear, and notice is given that copyins is by permission of ACM, Inc. To copy otherwise, to republish, t,o post on servers or to redistribute to lists, requires prior specific permission and/or a fee.

Rational Parametrizations of Algebraic Curves using a Canonical Divisor

Journal of Symbolic Computation, 1997

For an algebraic curve C with genus 0 the vector space L(D) where D is a divisor of degree 2 gives rise to a bijective morphism g from C to a conic C 2 in the projective plane. We present an algorithm that uses an integral basis for computing L(D) for a suitably chosen D. The advantage of an integral basis is that it contains all the necessary information about the singularities, so once the integral basis is known the L(D) algorithm does not need work with the singularities anymore. If the degree of C is odd, or more generally, if any odd degree rational divisor on C is known then we show how to construct a rational point on C 2 . In such cases a rational parametrization, which means de ned without algebraic extensions, of C 2 can be obtained. In the remaining cases a parametrization of C 2 de ned over a quadratic algebraic extension can be computed. A parametrization of C is obtained by composing the parametrization of C 2 with the inverse of the morphism g.