Dimensions Of Bivariate Spline Spaces Over Unconstricted Triangulations

The bivariate spline spaces S nr(?) on regular triangulations have very pop-ular use in many fields. Therefore, its dimension problem has been continuouslyconcerned by the mathematics and computer scholars. However, it is found thatthe dimensions of S nr(?) depend not only on the topological properties of trian-gulations, but also on the geometric shape. This makes it di?cult to determinethe dimensions. For the general regular triangulations, when n≥3r + 2 andn = 4, r = 1, the problem of dimensions has been solved.In this paper, we focus on the problem of dimensions of spline functionspace S nr(?) under the unconstricted triangulations and generalized unconstrictedtriangulations. The contents are arranged as follows. First, the dimensions ofbivariate spline function spaces S nr(?) under a class of special triangulations–unconstricted triangulations are achieved by studying the distribution of minimaldetermining set on star-type regional. Then, by introducing two new operatorsin triangulation construction, a kind of generalized unconstricted triangulation isdefined, which is an expansion of the unconstricted triangulations introduced byFarin in 2006. It contains some well-known triangulations, such as the Morgan-Scott and the Robbins triangulations, on which the dimension problems are dif-ficult to be solved. And then, the dimensions of bivariate C1 cubic spline spacesover generalized unconstricted triangulations is determined.