Cell reducing and the dimension of the C1 bivariate spline space

Abstract

In this paper, the problem of determining the dimension of the space S_n¹(△), n ≥ 3 of bivariate C¹ splines of degree ≤ n over a triangulation △ is considered. The piecewise polynomials are represented as blossoms, and the smoothness conditions are written as a system of linear equations. The rank of the system matrix is analysed by repeatedly reducing small subtriangulations (cells) at the boundary of a triangulation. It is shown that the dimension of the bivariate spline space S_n¹(△), n ≥ 3 is equal to Schumaker’s lower bound for a large class of triangulations.