Dimension of spline spaces with highest order smoothness over hierarchical T-meshes

Abstract

This paper discusses the dimension of spline spaces S(m,n,m−1,n−1,T) over a certain type of hierarchical T-mesh. The key step is to establish a bijection between the spline space S(m,n,m−1,n−1,T) and a univariate spline space defined in terms of the l-edges of the extended T-mesh with respect to bi-degree (m,n). We decompose the univariate spline space into an isomorphic direct sum using the theory of short exact sequences from homological algebra. Using this decomposition, we can obtain a formula for the dimension of the spline space S(m,n,m−1,n−1,T) over the required type of hierarchical T-mesh. We also construct a set of basis functions for the spline space.