The computation of simplicial approximations of implicitly defined two- dimensional manifolds
type: Item
instance of: scholarly article
title: The computation of simplicial approximations of implicitly defined two- dimensional manifolds
zbMATH Open document ID: 0815.65068
DOI: 10.1016/0898-1221(94)00175-8
author: M. L. Brodzik
author: Werner C. Rheinboldt
publication date: 1995-06-30T00:00:00Z
review text: Computation of a triangular approximation to the two-dimensional manifold \(M\) of solutions \(F(x)= 0\), where \(F: \mathbb{R}^ n\to \mathbb{R}^{n- 2}\), is explained. The described algorithm produces a data structure containing points approximately on the manifold, as well as adjacency information describing the triangles; it is asserted that manipulation of this structure is relatively efficient. This structure is generated with a procedure that is reminiscent of predictor-corrector methods for path following: Predictor points are first computed in tangent planes to the manifold, then a corrector step projects these points back onto the manifold. The tesselation proceeds as a ``frontal'' method; special techniques are described for filling out the tesselation uniformly in the neighborhood of a point on the manifold (essentially by subdividing the circular angle around the point), while care is taken to preserve the orientation of the manifold and its components. An adjustment technique is used when two fronts overlap (at seams). The paper contains a clear description of both the mathematical constructs and data structures. Although the paper reflects a FORTRAN 77 program, detailed pseudo-code adequate for implementation appears in the paper. Pictures corresponding to several numerical experiments, including an implicitly-defined sphere in \(\mathbb{R}^ 3\), the solution of equations describing deformation of a circular arc, and another special test problem, appear.
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 68U05
Mathematics Subject Classification ID: 65D18
Mathematics Subject Classification ID: 65D17
zbMATH DE Number: 701650
zbMATH Keywords: continuation methods
zbMATH Keywords: computer graphics
zbMATH Keywords: simplicial approximations
zbMATH Keywords: solution manifold
zbMATH Keywords: frontal method
zbMATH Keywords: triangular approximation
zbMATH Keywords: algorithm
zbMATH Keywords: predictor-corrector methods
zbMATH Keywords: path following
zbMATH Keywords: tesselation
zbMATH Keywords: numerical experiments
reviewed by: Ralph Baker Kearfott
describes a project that uses: PITCON
MaRDI profile type: Publication
full work available at URL: 0898-1221(94)00175-8
OpenAlex ID: W4213033603
cites work: Q3330398
cites work: An Algorithm for Piecewise-Linear Approximation of an Implicitly Defined Manifold
cites work: An Algorithm for Piecewise Linear Approximation of Implicitly Defined Two-Dimensional Surfaces
cites work: Q3798152
cites work: On the computation of multi-dimensional solution manifolds of parametrized equations
cites work: A new algorithm for two-dimensional numerical continuation
cites work: Q3881473
cites work: On the monotone convergence of Newton's method
cites work: Affine Invariant Convergence Theorems for Newton’s Method and Extensions to Related Methods
cites work: A non-linear finite element analysis of shallow circular arches
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 701650