Polynomial splines of non-uniform degree on triangulations: Combinatorial bounds on the dimension

Deepesh Toshniwal*, Thomas J.R. Hughes

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

3 Citations (Scopus)

Abstract

For T a planar triangulation, let Rm r(T) denote the space of bivariate splines on T such that f∈Rm r(T) is Cr(τ) smooth across an interior edge τ and, for triangle σ in T, f|σ is a polynomial of total degree at most m(σ)∈Z≥0. The map m:σ↦Z≥0 is called a non-uniform degree distribution on the triangles in T, and we consider the problem of computing (or estimating) the dimension of Rm r(T) in this paper. Using homological techniques, developed in the context of splines by Billera (1988), we provide combinatorial lower and upper bounds on the dimension of Rm r(T). When all polynomial degrees are sufficiently large, m(σ)≫0, we prove that the number of splines in Rm r(T) can be determined exactly. In the special case of a constant map m, the lower and upper bounds are equal to those provided by Mourrain and Villamizar (2013).

Original languageEnglish
Article number101763
Pages (from-to)1-22
Number of pages22
JournalComputer Aided Geometric Design
Volume75
DOIs
Publication statusPublished - 2019

Keywords

  • Dimension formula
  • Mixed polynomial degrees
  • Mixed smoothness
  • Splines
  • Triangulations

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