On degenerate surface patches
type: Item
instance of: scholarly article
title: On degenerate surface patches
zbMATH Open document ID: 0789.65006
DOI: 10.1007/BF02113892
author: V. Pereyra
published in: Numerical Algorithms
publication date: 1994-06-13T00:00:00Z
review text: The authors outline an algorithm to construct a geometrically \(C^ 1\)- surface through a set of given points with prescribed normals at that point. The problem naturally is that one has to start with a triangulation defined by the given points and the normal directions do not relate to any of the patches concurrent at the vertex. They solve the problem by using degenerate polynomial patches, i.e., patches constructed for coalescing base points. They note that the boundary curves are singular in that \(ds \to 0\) at the vertices and at the same time the curvature \(\to \infty\). Since the geometric theory of curvature presupposes nondegenerate \(ds\), one still does obtain a geometrically smooth curve on a geometrically smooth surface. (In fact, these curves have simple cusps at these points, as defined in algebraic geometry. In their example of the plane curve \(y = | x|^{2-\alpha}\), the curve is composed of two arcs fitting together smoothly at the origin; for \(x > 0\) the curve really represents a branch of \(y^ 2 = x^{4-2\alpha}\), with two branches forming a cusp at 0. Since the patches stop at the vertices, the other branch then is discarded).
Mathematics Subject Classification ID: 65D17
Mathematics Subject Classification ID: 53A04
zbMATH DE Number: 510229
zbMATH Keywords: degenerate surface patches
zbMATH Keywords: algorithm
zbMATH Keywords: degenerate polynomial patches
reviewed by: Heinrich Guggenheimer
MaRDI profile type: Publication
cites work: Q3975822
cites work: Degenerate polynomial patches of degree 11 for almost \(GC^ 2\) interpolation over triangles
cites work: Degenerate polynomial patches of degree 4 and 5 used for geometrically smooth interpolation in \(\mathbb{R}^ 3\)
cites work: Local smooth surface interpolation: A classification
full work available at URL: https://doi.org/10.1007/bf02113892
OpenAlex ID: W2088806350
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description: scientific article; zbMATH DE number 510229