Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness r and degree \(d\geq 4r+1\)
type: Item
instance of: scholarly article
title: Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness r and degree \(d\geq 4r+1\)
zbMATH Open document ID: 0668.41011
DOI: 10.1016/0167-8396(87)90028-8
author: Peter Alfeld
author: Bruce Piper
author: Larry L. Schumaker
published in: Computer Aided Geometric Design
publication date: 1987-01-01T00:00:00Z
review text: The paper considers bivariate splines. Let \(\Delta\) be a regular triangulation of a simply connected polygonal domain \(\Omega \subset {\mathbb{R}}^ 2\). Suppose the triangles of \(\Delta\) are labelled \(T^{[1]},...,T^{[M]}\) where M denotes the number of triangles. Given \(0\leq r\leq d\) we consider the linear space of bivariate splines \(S^ r_ d(\Delta)=\{s\in C^ r(\Omega):s|_{T^{[i]}}\in {\mathbb{P}}_ d\), \(i=1,...,M\}\), where \(P_ d\) is the \(\left( \begin{matrix} d+2\\ 2\end{matrix} \right)\)-dimensional linear space of bivariate polynomials of total degree d. Recently \textit{P. Alfeld} and \textit{L. L. Schumaker} [Constructing Apprxomation 3, 189-197 (1987; Zbl 0646.41008)] established the dimension of the spline spaces with \(d\geq 4r+1\), and \textit{J. Morgan} and \textit{R. Scott} in Math. Comp. 29, 736-740 (1975; Zbl 0307.65074) constructed local bases for the special case of r-1 and \(d\geq 5\). This work establishes the existence of minimally supported basis functions whenever \(d\geq 4r+1\). For the special cases \(r=1,2,3\) an explicit construction of a particular local basis is established. A fundamental tool in the analysis is the Bezier form of a bivariate polynomial.
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 4094088
zbMATH Keywords: bivariate splines
zbMATH Keywords: bivariate polynomials
zbMATH Keywords: minimally supported basis functions
reviewed by: M.-B. A. Babaev
MaRDI profile type: Publication
full work available at URL: 0167-8396(87)90028-8
OpenAlex ID: W2035789360
cites work: A trivariate Clough-Tocher scheme for tetrahedral data
cites work: Q3802004
cites work: The dimension of bivariate spline spaces of smoothness r for degree \(d\geq 4r+1\)
cites work: C1 quintic interpolation over triangles: Two explicit representations
cites work: A survey of curve and surface methods in CAGD
cites work: Q3716196
cites work: Q3732238
cites work: A Nodal Basis for C 1 Piecewise Polynomials of Degree n ≥5
cites work: Q3910883
cites work: Bounds on the dimension of spaces of multivariate piecewise polynomials
description: scientific article; zbMATH DE number 4094088