Details:
Title  Homological techniques for the analysis of the dimension of triangular spline spaces  Author(s)  Bernard Mourrain, Nelly Villamizar  Type  Article in Journal  Abstract  The 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.  Keywords  Splines, Triangulations, Bounds on the dimension, Powell–Sabin subdivisions  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717112001708 
Language  English  Journal  Journal of Symbolic Computation  Volume  50  Number  0  Pages  564  577  Year  2013  Edition  0  Translation 
No  Refereed 
No 
