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TitleHomological techniques for the analysis of the dimension of triangular spline spaces
Author(s) Bernard Mourrain, Nelly Villamizar
TypeArticle in Journal
AbstractThe spline space C_k^r(Δ) attached to a subdivided domain Δ of R^d is the vector space of functions of class C^r which are polynomials of degree ⩽k on each piece of this subdivision. Classical splines on planar rectangular grids play an important role in Computer Aided Geometric Design, and spline spaces over arbitrary subdivisions of planar domains are now considered for isogeometric analysis applications. We address the problem of determining the dimension of the space of bivariate splines C_k^r(Δ) for a triangulated region Δ in the plane. Using the homological introduced by Billera (1988), we number the vertices and establish a formula for an upper bound on the dimension. There is no restriction on the ordering and we obtain more accurate approximations to the dimension than previous methods and furthermore, in certain cases even an exact value can be found. The construction makes also possible to get a short proof for the dimension formula when k ⩾ 4 r + 1 , and the same method we use in this proof yields the dimension straightaway for many other cases.
KeywordsSplines, Triangulations, Bounds on the dimension, Powell–Sabin subdivisions
URL http://www.sciencedirect.com/science/article/pii/S0747717112001708
JournalJournal of Symbolic Computation
Pages564 - 577
Translation No
Refereed No