SPARSE PARAMETRIZATION OF PLANE CURVES (original) (raw)

Parametrization of algebraic curves defined by sparse equations

Applicable Algebra in Engineering, Communication and Computing, 2007

We present a new method for the rational parametrization of plane algebraic curves. The algorithm exploits the shape of the Newton polygon of the defining implicit equation and is based on methods of toric geometry.

Approximate parametrization of plane algebraic curves by linear systems of curves

Computer Aided Geometric Design, 2010

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 ǫ > 0 and an ǫ-irreducible algebraic affine plane curve C of proper degree d, we introduce the notion of ǫ-rationality, and we provide an algorithm to parametrize approximately affine ǫ-rational plane curves, without exact singularities at infinity, by means of linear systems of (d − 2)-degree curves. The algorithm outputs a rational parametrization of a rational curve C of degree at most 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.

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.

Computing parameterizations of rational algebraic curves

Proceedings of the international symposium on Symbolic and algebraic computation - ISSAC '94, 1994

In this paper I want to present a new method for computing parametrizations of algebraic curves. Basically this method is a direct application of integral basis computation. Examples show that this method is faster than older methods.

Parametrization of approximate algebraic curves by lines

Theoretical Computer Science, 2004

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).

Numerical Reparametrization of Rational Parametric Plane Curves

2013

In this paper, we present an algorithm for reparametrizing algebraic plane curves from a numerical point of view. That is, we deal with mathematical objects that are assumed to be given approximately. More precisely, given a tolerance ǫ > 0 and a rational parametrization P with perturbed float coefficients of a plane curve C, we present an algorithm that computes a parametrization Q of a new plane curve D such that Q is an ǫ-proper reparametrization of D. In addition, the error bound is carefully discussed and we present a formula that measures the "closeness" between the input curve C and the output curve D.

A New Algorithm for Implicitizing a Parametric Algebraic Surface

International Journal of Pure and Apllied Mathematics, 2015

Given a parametric representation of an algebraic projective surface S of the ordinary space we give a new algorithm for finding the implicit cartesian equation of S. The algorithm is based on finding a suitable finite number of points on S and computing, by linear algebra, the equation of the surface of least degree that passes through the points.

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.

Parametrization of approximate algebraic surfaces by lines

Computer Aided Geometric Design, 2005

SPARSE PARAMETRIZATION OF PLANE CURVES (original) (raw)

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