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

摘要

This paper discusses the dimension of spline spaces S(m, n, m - 1, n - 1, .J) 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, J) and a univariate spline space defined in terms of the 1-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, J) over the required type of hierarchical T-mesh. We also construct a set of basis functions for the spline space.