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On the dimension of spline spaces on planar T-meshes

Bernard Mourrain 1 
1 GALAAD2 - Géométrie , Algèbre, Algorithmes
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : We analyze the space of bivariate functions that are piecewise polynomial of bi-degree <= (m, m') and of smoothness r along the interior edges of a planar T-mesh. We give new combinatorial lower and upper bounds for the dimension of this space by exploiting homological techniques. We relate this dimension to the weight of the maximal interior segments of the T-mesh, defined for an ordering of these maximal interior segments. We show that the lower and upper bounds coincide, for high enough degrees or for hierarchical T-meshes which are enough regular. We give a rule of subdivision to construct hierarchical T-meshes for which these lower and upper bounds coincide. Finally, we illustrate these results by analyzing spline spaces of small degrees and smoothness.
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Submitted on : Sunday, September 13, 2015 - 9:32:33 AM
Last modification on : Saturday, June 25, 2022 - 7:40:49 PM
Long-term archiving on: : Tuesday, December 29, 2015 - 1:03:08 AM


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  • HAL Id : inria-00533187, version 3
  • ARXIV : 1011.1752



Bernard Mourrain. On the dimension of spline spaces on planar T-meshes. Mathematics of Computation, 2014, 83, pp.847-871. ⟨inria-00533187v3⟩



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