A univariate resultant-based implicitization algorithm for surfaces (original) (raw)

An Implicitization Algorithm for Rational Surfaces with no Base Points

Journal of Symbolic Computation, 2001

We present an implicitization algorithm which is free of extraneous factors if the rational parametric surface has no base points. This algorithm is based on the method of Sylvester for computing the resultant of three homogeneous polynomials in three variables. Some examples and computations illustrate the efficiency and limits of this method.

Implicitization of rational surfaces by means of polynomial interpolation

Computer Aided Geometric Design, 2002

A method for finding the implicit equation of a surface given by rational parametric equations is presented. The method is based on an efficient computation of the resultant by means of classical multivariate polynomial interpolation. The used approach considerably reduces the problem of intermediate expression swell and it can easily be implemented in parallel. 

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.

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.

Implicitizing rational surfaces with base points using the method of moving surfaces

Topics in Algebraic Geometry and Geometric Modeling, 2003

The method of moving planes and moving quadrics can express the implicit equation of a parametric surface as the determinant of a matrix M. The rows of M correspond to moving planes or moving quadrics that follow the parametric surface. Previous papers on the method of moving surfaces have shown that a simple base point has the effect of converting one moving quadric to a moving plane. A much more general version of the method of moving surfaces is presented in this paper that is capable of dealing with multiple base points. For example, a double base point has the effect (in this new version) of converting two moving quadrics into moving planes, eliminating one additional moving quadric, and eliminating a column of the matrix (i.e., a blending function of the moving surfaces)-thereby dropping the degree of the implicit equation by four. Furthermore, this is a unifying approach whereby tensor product surfaces, pure degree surfaces, and "corner-cut" surfaces, can all be implicitized under the same framework and do not need to be treated as distinct cases. The central idea in this approach is that if a surface has a base point of multiplicity k, the moving surface blending functions must have the same base point, but of multiplicity k − 1. Thus, we draw moving surface blending functions from the derivative ideal I , where I is the ideal of the parametric equations. We explain the general outline of the method and show how it works in some specific cases. The paper concludes with a discussion of the method from the point of view of commutative algebra.

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.

Residue and Implicitization Problem for Rational Surfaces

Applicable Algebra in Engineering, Communication and Computing, 2004

The implicitization problem of rational surfaces is a central challenge in computational aided geometric design. We propose a new algorithm to solve it, based on the residue calculus in the general multivariate setting. The proposed approach allows us to treat surfaces with base points (without geometric hypothesis on the zero-locus of base points). The eciency of the method is illustrated by some examples computed with maple software. We also give another application of the residue calculus to the computation of osets of rational parametric surfaces.

Implicitization of Hypersurfaces

We present new, practical algorithms for the hypersurface implicitization problem: namely, given a parametric description (in terms of polynomials or rational functions) of the hypersurface, find its implicit equation. Two of them are for polynomial parametrizations: one algorithm, " ElimTH " , has as main step the computation of an elimination ideal via a truncated, homogeneous Gröbner basis. The other algorithm, " Direct " , computes the implicitization directly using an approach inspired by the generalized Buchberger-Möller algorithm. Either may be used inside the third algorithm, " RatPar " , to deal with parametrizations by rational functions. Finally we show how these algorithms can be used in a modular approach, algorithm " ModImplicit " , for avoiding the high costs of arithmetic with rational numbers. We exhibit experimental timings to show the practical efficiency of our new algorithms.

Resultants, Implicit Parameterizations, and Intersections of Surfaces

Mathematical Software – ICMS 2018 6th International Conference, South Bend, IN, USA, July 24-27, 2018, Proceedings, 2018

A fundamental problem in computer graphics and computer aided design is to convert between a parameterization of a surface and an implicit representation of it. Almost as fundamental is to derive a parameterization for the intersection of two surfaces. In these problems, it seems that resultants, specifically the Dixon resultant , have been underappreciated. Indeed, several well known papers from ten to twenty years ago reported unsuitability of resultant techniques. To the contrary, we show that the Dixon resultant is an extremely effective and efficient method to compute an implicit representation. To use resultants to compute a parameterization of an intersection, we introduce the concept of an "implicit parameterization." Unlike the conventional parameterization of a curve where x, y, and z are each explicitly given as functions of, say, t, we have three implicit functions, one each for (x, t), (y, t), and (z, t). This concept has rarely been mentioned before. We show that given a (conventional) parameterization for one surface and either an implicit equation for the second, or a parameterization for it, it is straightforward to compute an implicit parameterization for the intersection. Doing so is very easy for the Dixon resultant, but can be very daunting even for well respected Gröbner bases programs. Further, we demonstrate that such implicit parameterizations are useful. We use builtin 3D plotting utilities of a computer algebra system to graph the intersection using our implicit parameterization. We do this for examples that are more complex than the quadric examples usually discussed in intersection papers.

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.