Abstract
For T a planar triangulation, let Rmr(T) denote the space of bivariate splines on T such that f∈Rmr(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 Rmr(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 Rmr(T). When all polynomial degrees are sufficiently large, m(σ)≫0, we prove that the number of splines in Rmr(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).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.