Rational quadratic approximation to real algebraic curves

doi:10.1016/j.cagd.2004.07.009

Computer Aided Geometric Design

Volume 21, Issue 8, October 2004, Pages 805-828

https://doi.org/10.1016/j.cagd.2004.07.009 Get rights and content

Abstract

An algorithm is proposed to give a global approximation of an implicit real plane algebraic curve with rational quadratic B-spline curves. The algorithm consists of four steps: topology determination, curve segmentation, segment approximation and curve tracing. Due to the detailed geometric analysis, high accuracy of approximation may be achieved with a small number of quadratic segments. The final approximation keeps many important geometric features of the original curve such as the topology, convexity and sharp points. Our method is implemented and experiments show that it may achieve better approximation bound with less segments than previously known methods. We also extend the method to approximate spatial algebraic curves.

References (25)

S. Abhyankar et al.
Automatic parameterization of rational curves and surfaces, III: algebraic plane curves
Computer Aided Geometric Design
(1988)
C. Bajaj et al.
Tracing surface intersections
Computer Aided Geometric Design
(1988)
F.L. Chen et al.
Computing real inflection points of cubic algebraic curves
Computer Aided Geometric Design
(2003)
R. Farouki
Hierarchical segmentations of algebraic curves and some applications
L. Gonzalez-Vega et al.
Efficient topology determination of implicitly defined algebraic plane curves
Computer Aided Geometric Design
(2002)
H. Hong
An efficient method for analyzing the topology of plane real algebraic curves
Math. Comput. Simulation
(1996)
L. Piegl et al.
Curve and surface constructions using rational B-spline
Computer-Aided Design
(1987)
H.P. Yang et al.
Control points adjustment for B-spline curve approximation
Computer-Aided Design
(2004)
C. Bajaj et al.
Piecewise rational approximations of real algebraic curves
J. Comput. Math.
(1997)
J. Bondy et al.
Graph Theory with Applications
(1976)

G.Z. Chang et al.

Over and Over Again

(1997)

F.L. Chen et al.

Interval parametrization of planar algebraic curves

Cited by (25)

Globally certified G<sup>1</sup> approximation of planar algebraic curves
2024, Journal of Computational and Applied Mathematics
Topology analysis and geometric feature determination for curves and surfaces are basic problems in geometric modeling and computer aided design. When the curves and surfaces are given in implicit expressions, the problems become more difficult. This paper presents a complete algorithm framework for the problem of certified $G^{1}$ approximation of a planar algebraic curve under global error control. Compared to other work on topology and geometric analysis of algebraic curves, this method presents a new topology analysis as well as a geometric information computation. Moreover, this method uses generalized asymptotes as the approximation of the unbounded part, which makes it achieve global approximation with given precision.
Numerical proper reparametrization of parametric plane curves
2015, Journal of Computational and Applied Mathematics
We present an algorithm for reparametrizing algebraic plane curves from a numerical point of view. More precisely, given a tolerance $ϵ > 0$ and a rational parametrization $P$ of a plane curve $C$ with perturbed float coefficients, we present an algorithm that computes a parametrization $Q$ of a new plane curve $D$ such that $Q$ is an $ϵ$ –proper reparametrization of $D$ . In addition, the error bound is carefully discussed and we present a formula that measures the “closeness” between the input curve $C$ and the output curve $D$ .
Certified rational parametric approximation of real algebraic space curves with local generic position method
2013, Journal of Symbolic Computation
Citation Excerpt :
It works well for low degree algebraic space curves. In Gao and Li (2004), the authors presented an algorithm to approximate an irreducible space curves under a given precision. It is based on the fact that there exists a birational map between the projection curve for some direction and the irreducible algebraic space curve.
In this paper, an algorithm is given for determining the topology of an algebraic space curve and to compute a certified $G^{1}$ rational parametric approximation of the algebraic space curve. The algorithm works by extending to dimension one the local generic position method for solving zero-dimensional polynomial equation systems. Here, certified means that the approximation curve and the original curve have the same topology and their Hausdorff distance is smaller than a given precision. The main advantage of the algorithm, inherited from the local generic position method, is that the topology computation and approximation for a space curve are directly reduced to the same tasks for two plane curves. In particular, an error bound of the approximation space curve is deduced explicitly from the error bounds of the approximation plane curves. The complexity of the algorithm is also analyzed. Its effectivity is shown on some non-trivial examples.
A symbolic-numerical approach to approximate parameterizations of space curves using graphs of critical points
2013, Journal of Computational and Applied Mathematics
Citation Excerpt :
Hence, suitable techniques producing (only) approximate parameterizations are often used to avoid these problems. Various related results for planar curves exist, cf. [15–19]. In this paper we will focus on the not very often discussed case of space algebraic curves which are defined as the intersections of algebraic surfaces.
A simple algorithm for computing an approximate parameterization of real space algebraic curves using their graphs of critical points is designed and studied in this paper. The first step is determining a suitable space graph which contains all critical points of a real algebraic space curve $C$ implicitly defined as the complete intersection of two surfaces. The construction of this graph is based on one projection of $C$ in a general position onto an $x y$ -plane and on an intentional choice of vertices. The second part of the designed method is a computation of a spline curve which replaces the edges of the constructed graph by segments of a chosen free-form curve. This step is formulated as an optimization problem when the objective function approximates the integral of the squared Euclidean distance of the constructed approximate curve to the intersection curve. The presented method, based on combining symbolic and numerical steps to the approximation problem, provides approximate parameterizations of space algebraic curves from a small number of approximating arcs. It may serve as a first step to several problems originating in technical practice where approximation curve parameterizations are needed.
Parallel computation of real solving bivariate polynomial systems by zero-matching method
2013, Applied Mathematics and Computation
We present a new algorithm for solving the real roots of a bivariate polynomial system $Σ = {f (x, y), g (x, y)}$ with a finite number of solutions by using a zero-matching method. The method is based on a lower bound for the bivariate polynomial system when the system is non-zero. Moreover, the multiplicities of the roots of $Σ = 0$ can be obtained by the associated quotient ring technique and a given neighborhood. From this approach, the parallelization of the method arises naturally. By using a multidimensional matching method this principle can be generalized to the multivariate equation systems.
Certified approximation of parametric space curves with cubic B-spline curves
2012, Computer Aided Geometric Design
Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given precision and preserve the geometric features of the given curve such as the topology, singular points, etc. The approximated curve is divided into segments called quasi-cubic Bézier curve segments which have properties similar to a cubic rational Bézier curve. And the approximate curve is naturally constructed as the associated cubic rational Bézier curve of the control tetrahedron of a quasi-cubic curve. A novel optimization method is proposed to select proper weights in the cubic rational Bézier curve to approximate the given curve. The error of the approximation is controlled by the size of its tetrahedron, which converges to zero by subdividing the curve segments. As an application, approximate implicit equations of the approximated curves can be computed. Experiments show that the method can approximate space curves of high degrees with high precision and very few cubic Bézier curve segments.

View all citing articles on Scopus

^☆: Partially supported by a NKBR Project of China and US NSF grant CCR-0201253.

View full text

Rational quadratic approximation to real algebraic curves☆

Abstract

Computer Aided Geometric Design

Computer Aided Geometric Design

Computer Aided Geometric Design

Computer Aided Geometric Design

Math. Comput. Simulation

Computer-Aided Design

Computer-Aided Design

Piecewise rational approximations of real algebraic curves

J. Comput. Math.

Graph Theory with Applications

Over and Over Again

Interval parametrization of planar algebraic curves