Approximate Implicitization Using Monoid Curves and Surfaces (original) (raw)

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.

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. 

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.

Comparative benchmarking of methods for approximate implicitization

Geometric Design and …, 2004

Recently a variety of algorithms for the approximation of parametric curves and surfaces by algebraic representations have been developed. We test three of these methods on several test cases, keeping track of time and memory consumption of the implementations and the quality of the approximation. Additionally we discuss some qualitative aspects of the different methods. XXX 1 xxx and xxx (eds.), pp. 1-4.

Approximate implicitization of planar curves by piecewise rational approximation of the distance function

Applicable Algebra in Engineering, Communication and Computing, 2006

We present an approximate implicitization method for planar curves. The computed implicit representation is a piecewise rational approximation of the distance function to the given parametric curve. The proposed method consists of four main steps: quadratic B-spline approximation of the given parametric curve, data reduction, segments-wise implicitization, multiplying with suitable polynomial factors. These segments are joined such that the collection generate a global C r spline function which approximates the distance function, for r = 0, 1.

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.

Approximate algebraic methods for curves and surfaces and their applications

Proceedings of the 21st spring conference on Computer graphics - SCCG '05, 2005

We report on approximate techniques for conversion between the implicit and the parametric representation of curves and surfaces, i.e., implicitization and parameterization. It is shown that these techniques are able to handle general free-form surfaces, and they can therefore be used to exploit the duality of implicit and parametric representations. In addition, we discuss several applications of these techniques, such as detection of self-intersections, raytracing, footpoint computation and parameterization of scattered data for parametric curve or surface fitting.

Implicitization of curves and (hyper) surfaces using predicted support

We reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation. For predicting the implicit support, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory. Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial. We implement our methods on Maple, and some on Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces. We apply our approach to approximate implicitization, and quantify the accuracy of the approximate output, which turns out to be satisfactory on all tested examples; we also relate our measures to Hausdorff distance. In building a square or rectangular matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complexes offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice. We compare our prototype to existing software and find that it is rather competitive.

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.