TY - JOUR
T1 - Polynomial splines of non-uniform degree on triangulations
T2 - Combinatorial bounds on the dimension
AU - Toshniwal, Deepesh
AU - Hughes, Thomas J.R.
PY - 2019
Y1 - 2019
N2 - For T a planar triangulation, let Rm r(T) denote the space of bivariate splines on T such that f∈Rm r(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 Rm 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 Rm r(T). When all polynomial degrees are sufficiently large, m(σ)≫0, we prove that the number of splines in Rm 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).
AB - For T a planar triangulation, let Rm r(T) denote the space of bivariate splines on T such that f∈Rm r(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 Rm 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 Rm r(T). When all polynomial degrees are sufficiently large, m(σ)≫0, we prove that the number of splines in Rm 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).
KW - Dimension formula
KW - Mixed polynomial degrees
KW - Mixed smoothness
KW - Splines
KW - Triangulations
UR - http://www.scopus.com/inward/record.url?scp=85072870193&partnerID=8YFLogxK
U2 - 10.1016/j.cagd.2019.07.002
DO - 10.1016/j.cagd.2019.07.002
M3 - Article
AN - SCOPUS:85072870193
SN - 0167-8396
VL - 75
SP - 1
EP - 22
JO - Computer Aided Geometric Design
JF - Computer Aided Geometric Design
M1 - 101763
ER -