On the equation of a parametric algebraic surface (original) (raw)

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.

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-...

First steps towards radical parametrization of algebraic surfaces

Computer Aided Geometric Design, 2013

We introduce the notion of radical parametrization of a surface, and we provide algorithms to compute such type of parametrizations for families of surfaces, like: Fermat surfaces, surfaces with a high multiplicity (at least the degree minus 4) singularity, all irreducible surfaces of degree at most 5, all irreducible singular surfaces of degree 6, and surfaces containing a pencil of low-genus curves. In addition, we prove that radical parametrizations are preserved under certain type of geometric constructions that include offset and conchoids.

Rational Parametrizations of Real Cubic Surfaces

1995

Real cubic algebraic surfaces may be described by either implicit or parametric equations. Each of these representations has strengths and weaknesses and have been used extensively in computer graphics. Applications involving both representations include the efficient computation of surface intersections, and triangulation of curved surfaces. One particularly useful representation is the rational parametrization, where the three spatial coordinates are given by rational functions of two parameters. Rational parametrizations speed up many computations, and their relatively simple structure allows one to control and avoid singularities in the parametrization. These parametrizations take on different forms for different classes of cubic surfaces. Classification of real cubic algebraic surfaces into five families for the nonsingular case is based on the configuration of twentyseven lines on them. We provide a method of extracting all these lines and from there a rational parametrization of each of these families. The parametrizations of the real cubic surface components are constructed using a pair of real skew lines for those three families which have them, and remarkably using a complex conjugate pair of skew lines, in a fourth family. The parametrization is based on the fact that a real line generally intersects a cubic surface at three points. Points on the surface are obtained by intersecting the surface with lines that pass through points on the two skew lines. We also analyze the image of the derived rational parametrization for both real and complex parameter values, together with "base" points where the parametrizations are ill-defined.

Rational parametrizations of nonsingular real cubic surfaces

ACM Transactions on Graphics, 1998

Real cubic algebraic surfaces may be described by either implicit or parametric equations. Each of these representations has strengths and weaknesses and have been used extensively in computer graphics. Applications involving both representations include the e cient computation of surface intersections, and triangulation of curved surfaces. One particularly useful representation is the rational parametrization, where the three spatial coordinates are given by rational functions of two parameters. Rational parametrizations speed up many computations, and their relatively simple structure allows one to control and avoid singularities in the parametrization. These parametrizations take on di erent forms for di erent classes of cubic surfaces. Classi cation of real cubic algebraic surfaces into ve families for the nonsingular case is based on the con guration of twentyseven lines on them. We provide a method of extracting all these lines and from there a rational parametrization of each of these families. The parametrizations of the real cubic surface components are constructed using a pair of real skew lines for those three families which have them, and remarkably using a complex conjugate pair of skew lines, in a fourth family. The parametrization is based on the fact that a real line generally intersects a cubic surface at three points. Points on the surface are obtained by intersecting the surface with lines that pass through points on the two skew lines. We also analyze the image of the derived rational parametrization for both real and complex parameter values, together with \base" points where the parametrizations are ill-de ned.

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).

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.

Implicit representation of rational parametric surfaces

1992

In many applications we need to compute the implicit representation of rational parametric surfaces. Previously, resultants and Gröbner bases have been applied to this problem. However, these techniques at times result in an extraneous factors along with the implicit equation and fail altogether when a parametrization has base points. In this paper we present algorithms to implicitize rational parametric surfaces with and without base points.

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.

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.