An Implicitization Algorithm for Rational Surfaces with no Base Points (original) (raw)
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.
Computer Aided Geometric Design, 2013
The availability of the implicit equation of a plane curve or of a 3D surface can be very useful in order to solve many geometric problems involving the considered curve or surface: for example, when dealing with the point position problem or answering intersection questions. On the other hand, it is well known that in most cases, even for moderate degrees, the implicit equation is either difficult to compute or, if computed, the high degree and the big size of the coefficients makes extremely difficult its use in practice. We will show that, for several problems involving plane curves, 3D surfaces and some of their constructions (for example, offsets), it is possible to use the implicit equation (or, more precisely, its properties) without needing to explicitly determine it. We replace the computation of the implicit equation by the evaluation of the considered parameterizations in a set of points and its use, in order to deal with the considered geometric problems, is translated into one or several generalized eigenvalue problems on matrix pencils (depending again on several evaluations of the considered parameterizations). This is the so called "Polynomial Algebra by Values" approach where the huge polynomial equations coming from Elimination Theory (e.g., using resultants) are replaced by big structured and sparse numerical matrices. For these matrices there are well known numerical techniques allowing to provide the results we need to answer the geometric questions on the considered curves and surfaces.
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.
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.
An implicitization algorithm with fewer variables
Computer Aided Geometric Design, 1995
In this paper we present a general implicitization algorithm for rational parameterizations, using Gr6bner Bases, which: (i) is valid for general parametric varieties (i.e. allowing both rational or polynomial parameterizations), (ii) computes the greatest ideal of polynomials vanishing over the variety and (iii) uses only as many variables as the number of parameters plus coordinates. We give examples of the performance of our algorithm in the CoCoa system, comparing the obtained results with other algorithms. * Corresponding author. Emaih gutierrez@ccucvx.unican.es. 1 Partially supported by CICyT-PB 89/0379/C02/01 and EPRIT Bra-POSSO, 6846. 0167-8396/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSD10167-8396(94)0001 1-G
Implicitizing Rational Curves by the Method of Moving Algebraic Curves
Journal of Symbolic Computation, 1997
A function F (x, y, t) that assigns to each parameter t an algebraic curve F (x, y, t) = 0 is called a moving curve. A moving curve F (x, y, t) is said to follow a rational curve x = x(t)/w(t), y = y(t)/w(t) if F (x(t)/w(t), y(t)/w(t), t) is identically zero. A new technique for finding the implicit equation of a rational curve based on the notion of moving conics that follow the curve is investigated. For rational curves of degree 2n with no base points the method of moving conics generates the implicit equation as the determinant of an n × n matrix, where each entry is a quadratic polynomial in x and y, whereas standard resultant methods generate the implicit equation as the determinant of a 2n × 2n matrix where each entry is a linear polynomial in x and y. Thus implicitization using moving conics yields more compact representations for the implicit equation than standard resultant techniques, and these compressed expressions may lead to faster evaluation algorithms. Moreover whereas resultants fail in the presence of base points, the method of moving conics actually simplifies, because when base points are present some of the moving conics reduce to moving lines.
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.
Using polynomial interpolation for implicitizing algebraic curves
Computer Aided Geometric Design, 2001
A simple algorithm for finding the implicit equation of a rational plane algebraic curve given by its parametric equations is presented. The algorithm is based on an efficient computation of the resultant by means of classical bivariate polynomial interpolation. One of the main features of the used approach is the fact that it considerably reduces the problem of intermediate expression swell, which is usually present in many computer algebra algorithms.
Approximate Implicitization Using Linear Algebra
Journal of Applied Mathematics, 2012
We consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore well suited to floating-point implementation in computer-aided geometric design (CAGD) systems. We unify the approaches under the names of commonly known polynomial basis functions and consider various theoretical and practical aspects of the algorithms. We offer new methods for a least squares approach to approximate implicitization using orthogonal polynomials, which tend to be faster and more numerically stable than some existing algorithms. We propose several simple propositions relating the properties of the polynomial bases to their implicit approximation properties.