Dimensions of Biquadratic and Bicubic Spline Spaces over Hierarchical T-meshes

TL;DR

This paper derives concise dimension formulas for biquadratic and bicubic spline spaces over hierarchical T-meshes, improving understanding and reducing restrictions compared to previous methods.

Contribution

It introduces new, simplified dimension formulas for biquadratic C1 and bicubic C2 spline spaces over hierarchical T-meshes, including a new type of T-mesh.

Findings

Dimension formula for biquadratic C1 spline spaces derived

Dimension formula for bicubic C2 spline spaces with fewer restrictions

New dimension formula for bicubic C2 spline spaces over a novel hierarchical T-mesh

Abstract

This paper discusses the dimensions of biquadratic C1 spline spaces and bicubic C2 spline spaces over hierarchical T-meshes using the smoothing cofactor-conformality method. We obtain the dimension formula of biquadratic C1 spline spaces over hierarchical T-meshes in a concise way. In addition, we provide a dimension formula for bicubic C2 spline spaces over hierarchical T-mesh with fewer restrictions than that in the previous literature. A dimension formula for bicubic C2 spline spaces over a new type hierarchical T-mesh is also provided.