Dimension of Bi-degree $(d,d)$ Spline Spaces with the Highest Order of Smoothness over Hierarchical T-Meshes

TL;DR

This paper derives a formula for the dimension of bi-degree (d,d) spline spaces with maximum smoothness over hierarchical T-meshes, using the smoothing cofactor-conformality method and recursive calculations.

Contribution

It introduces a recursive method to compute the dimension of spline spaces over hierarchical T-meshes and proposes a mesh modification strategy for stability.

Findings

Derived a dimensional formula for spline spaces over hierarchical T-meshes.

Proposed a mesh modification strategy to ensure dimension stability.

Established equivalence of spline space dimension to that of a lower-degree space over the CVR graph.

Abstract

In this article, we study the dimension of the spline space of di-degree $(d,d)$ with the highest order of smoothness over a hierarchical T-mesh $\mathscr T$ using the smoothing cofactor-conformality method. Firstly, we obtain a dimensional formula for the conformality vector space over a tensor product T-connected component. Then, we prove that the dimension of the conformality vector space over a T-connected component of a hierarchical T-mesh under the tensor product subdivision can be calculated in a recursive manner. Combining these two aspects, we obtain a dimensional formula for the bi-degree $(d,d)$ spline space with the highest order of smoothness over a hierarchical T-mesh $\mathscr T$ with mild assumption. Additionally, we provide a strategy to modify an arbitrary hierarchical T-mesh such that the dimension of the bi-degree $(d,d)$ spline space is stable over the modified hierarchical T-mesh. Finally, we prove that the dimension of the spline space over such a hierarchical T-mesh is the same as that of a lower-degree spline space over its CVR graph. Thus, the proposed solution can pave the way for the subsequent construction of basis functions for spline space over such a hierarchical T-mesh.