Representing rational curve segments and surface patches using semi-algebraic sets
type: Item
instance of: scholarly article
title: Representing rational curve segments and surface patches using semi-algebraic sets
zbMATH Open document ID: 1439.65024
author: Li-Yong Shen
author: Sonia Pérez Díaz
author: Yifei Feng
author: Ron Goldman
published in: Computer Aided Geometric Design
publication date: 2019-11-27T00:00:00Z
full work available at URL: http://hdl.handle.net/10017/41550
review text: In the work for generating surfaces by (piecewise) rational curves or surfaces, the representation of the surfaces (or curves) using semi-algebraic sets is an important tool. Such surfaces have many useful applications and there are also other methods to define them, such as radial basis functions. In this particular approach, rational curves or surfaces defined by tensor-products of rational curves are used. The main idea is the generation of globally defined functions whose local parts (segments) the desired surfaces are (in other words, the parts of the surfaces we really need are sections of the unbounded ones). They are defined implicitly by semi-algebraic sets, and the provided algorithms for computing the surfaces work well unless there are singularities in the rational function. These singularities need to be identified, and this is achieved by creating boxes about the singularities where the curves are generated within a certain numerical error (by contrast, there is no error if there is no singularity). Many numerical examples are provided to show the success of the methods.
Mathematics Subject Classification ID: 65D10
Mathematics Subject Classification ID: 65D17
Mathematics Subject Classification ID: 65D12
zbMATH DE Number: 7137425
zbMATH Keywords: rational curve segment
zbMATH Keywords: rational surface patch
zbMATH Keywords: semi-algebraic set
zbMATH Keywords: implicitization
reviewed by: Martin D. Buhmann
describes a project that uses: Implicit
MaRDI profile type: Publication
OpenAlex ID: W2969297921
cites work: Selfintersection of composite curves and surfaces
cites work: Q3714165
cites work: Radial Basis Functions
cites work: Quantifier elimination and cylindrical algebraic decomposition. Proceedings of a symposium, Linz, Austria, October 6--8, 1993
cites work: The \(\mu \)-basis and implicitization of a rational parametric surface
cites work: Theoretically Based Robust Algorithms for Tracking Intersection Curves of Two Deforming Parametric Surfaces
cites work: The μ-basis of a planar rational curve—properties and computation
cites work: A generic position based method for real root isolation of zero-dimensional polynomial systems
cites work: Rectangular corner cutting and Dixon \({\mathcal A}\)-resultants
cites work: Q5301664
cites work: Residue and implicitization problem for rational surfaces
cites work: Q3424927
cites work: \(\mu \)-bases and singularities of rational planar curves
cites work: Algorithm for implicitizing rational parametric surfaces
cites work: Subdivision methods for solving polynomial equations
cites work: Computation of the singularities of parametric plane curves
cites work: Computing the singularities of rational surfaces
cites work: Computing real roots of real polynomials
cites work: Homeomorphic approximation of the intersection curve of two rational surfaces
cites work: Implicitization using univariate resultants
cites work: Finding the number of roots of a polynomial in a plane region using the winding number
cites work: Geometric conditions of non-self-intersecting NURBS surfaces
DOI: 10.1016/J.CAGD.2019.101770
description: scientific article; zbMATH DE number 7137425