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 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 T
Nearby in the stack
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.