Polynomial splines of non-uniform degree on triangulations: Combinatorial bounds on the dimension
•We study spaces of bivariate polynomial splines on planar triangulations.•The splines are composed of pieces with different polynomial degrees on different triangles.•We provide combinatorial upper and lower bounds on the dimension of such splines.•Stability of the dimension in high degree is shown...
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| Vydáno v: | Computer aided geometric design Ročník 75; s. 101763 |
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| Jazyk: | angličtina |
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Elsevier B.V
01.11.2019
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| ISSN: | 0167-8396, 1879-2332 |
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| Abstract | •We study spaces of bivariate polynomial splines on planar triangulations.•The splines are composed of pieces with different polynomial degrees on different triangles.•We provide combinatorial upper and lower bounds on the dimension of such splines.•Stability of the dimension in high degree is shown.•Example applications are presented; accompanying Macaulay2 package is publicly available.
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). |
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| AbstractList | •We study spaces of bivariate polynomial splines on planar triangulations.•The splines are composed of pieces with different polynomial degrees on different triangles.•We provide combinatorial upper and lower bounds on the dimension of such splines.•Stability of the dimension in high degree is shown.•Example applications are presented; accompanying Macaulay2 package is publicly available.
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). |
| ArticleNumber | 101763 |
| Author | Toshniwal, Deepesh Hughes, Thomas J.R. |
| Author_xml | – sequence: 1 givenname: Deepesh surname: Toshniwal fullname: Toshniwal, Deepesh email: d.toshniwal@tudelft.nl organization: Delft Institute of Applied Mathematics, Delft University of Technology, the Netherlands – sequence: 2 givenname: Thomas J.R. surname: Hughes fullname: Hughes, Thomas J.R. organization: Oden Institute for Computational Engineering and Sciences, University of Texas at Austin, USA |
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| Cites_doi | 10.1216/RMJ-1984-14-1-251 10.1016/j.cagd.2012.03.025 10.1007/s00365-017-9367-5 10.1016/j.jsc.2012.10.002 10.1016/j.cma.2004.10.008 10.1016/j.automatica.2009.09.017 10.1016/j.cagd.2015.11.001 10.1006/aama.1997.0534 10.1016/j.cma.2016.11.009 10.1007/s11390-012-1268-2 10.1007/BF01890563 10.1006/jabr.1997.7361 10.1016/S0022-4049(97)00026-1 10.1145/882262.882295 10.1016/S0167-8396(03)00096-7 10.1007/BF02574678 10.1006/aama.1997.0533 10.1007/BF01386434 10.1016/j.cagd.2012.12.005 10.1090/S0002-9947-1988-0965757-9 |
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