### Dimension Formulae for Tensor-Product Spline Spaces with Homogeneous Boundary Conditions over Regular T-meshes

#### Abstract

A regular T-mesh is basically a rectangular grid that allows T-junctions over a rectangular domain. In this paper, we mainly study the dimension of

bivariate tensor-product spline space $S_{m,n}^{\alpha,\beta}(\mathcal

{T};\alpha_{1},\alpha_{2},\beta_{1},\beta_{2})$ with homogeneous boundary conditions over a regular T-mesh $\mathcal {T}$. By using B-net method, we construct a minimal determining set for $S_{m,n}^{\alpha,\beta}(\mathcal{T};\alpha_{1},\alpha_{2},\beta_{1},\beta_{2})$ by removing some unwanted domain points from the minimal determining set for $S_{m,n}^{\alpha,\beta}(\mathcal {T})$ given by Deng et al.. The new results are useful in the fields of computer aided geometric design, such as surface approximation, model design, and so on.

bivariate tensor-product spline space $S_{m,n}^{\alpha,\beta}(\mathcal

{T};\alpha_{1},\alpha_{2},\beta_{1},\beta_{2})$ with homogeneous boundary conditions over a regular T-mesh $\mathcal {T}$. By using B-net method, we construct a minimal determining set for $S_{m,n}^{\alpha,\beta}(\mathcal{T};\alpha_{1},\alpha_{2},\beta_{1},\beta_{2})$ by removing some unwanted domain points from the minimal determining set for $S_{m,n}^{\alpha,\beta}(\mathcal {T})$ given by Deng et al.. The new results are useful in the fields of computer aided geometric design, such as surface approximation, model design, and so on.

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