## Abstract

For T a planar triangulation, let R_{m} ^{r}(T) denote the space of bivariate splines on T such that f∈R_{m} ^{r}(T) is C^{r(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 R_{m} ^{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 R_{m} ^{r}(T). When all polynomial degrees are sufficiently large, m(σ)≫0, we prove that the number of splines in R_{m} ^{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 language | English |
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Article number | 101763 |

Pages (from-to) | 1-22 |

Number of pages | 22 |

Journal | Computer Aided Geometric Design |

Volume | 75 |

DOIs | |

Publication status | Published - 2019 |

## Keywords

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