Dimensions of bivariate C~1 cubic spline spaces over unconstricted triangulations with valence six

摘要

In this paper, we consider the open problem of dimensions of bivariate C~1 cubic spline spaces. Firstly, by introducing two new operators, element-3 and element-4, in constructing a triangulation, a kind of unconstricted triangulations with valence 6 is defined, which is an extension of the unconstricted triangulation introduced by Farin in [Dimensions of spline spaces over unconstricted triangulations, J. Comput. Appl. Math., 192(2006), pp.320-327] and some well-known triangulations such as the Morgan-Scott triangulation and the Robbins triangulation are therefore included. Then, by using the technique of minimal determining set, the dimension of bivariate C~1 cubic spline spaces over the unconstricted triangulation with valence six is determined and the Lagrange interpolation on all vertices in the unconstricted triangulation with valence 6 is considered.