The mu-basis of a rational ruled surface (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.

A Direct Approach to Computing theμ-basis of Planar Rational Curves

Journal of Symbolic Computation, 2001

This paper presents an O(n 2) algorithm, based on Gröbner basis techniques, to compute the µ-basis of a degree n planar rational curve. The prior method involved solving a set of linear equations whose complexity by standard numerical methods was O(n 3). The µ-basis is useful in computing the implicit equation of a parametric curve and can express the implicit equation in the form of a determinant that is smaller than that obtained by taking the resultant of the parametric equations.

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. 

Computing self-intersection curves of rational ruled surfaces

Computer Aided Geometric Design, 2009

An algorithm is presented to compute the self-intersection curves of a rational ruled surface based on the theory of μ-bases. The algorithm starts by constructing the principal subresultants for a μ-basis of the rational ruled surface. The principal subresultant coefficients provide information about not only the parametric loci of the self-intersection curves, but also the orders of the self-intersection curves. Based on this observation, an efficient algorithm is provided to compute the parametric loci of the self-intersection curves as well as their corresponding orders. The isolated singular points of the rational ruled surface are also computed.

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.

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.

A univariate resultant-based implicitization algorithm for surfaces

Journal of Symbolic Computation, 2008

In this paper, we present a new algorithm for computing the implicit equation of a rational surface V from a rational parametrization P(t). The algorithm is valid independent of the existence of base points, and is based on the computation of polynomial gcds and univariate resultants. Moreover, we prove that the resultant-based formula provides a power of the implicit equation. In addition, performing a suitable linear change of parameters, we prove that this power is indeed the degree of the rational map induced by the parametrization. We also present formulas for computing the partial degrees of the implicit equation.

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.