Space Mapping of Spline Spaces over Hierarchical T-meshes

TL;DR

This paper establishes a bijective, isomorphic mapping between biquadratic spline spaces over hierarchical T-meshes and piecewise constant spaces over CVR graphs, enabling efficient surface fitting.

Contribution

It introduces a novel structure and method for constructing basis functions of biquadratic spline spaces over hierarchical T-meshes, with proven properties and practical applications.

Findings

The basis functions are linearly independent and complete.

The basis functions satisfy the partition of unity.

Effective surface fitting demonstrates the basis functions' efficiency.

Abstract

In this paper, we construct a bijective mapping between a biquadratic spline space over the hierarchical T-mesh and the piecewise constant space over the corresponding crossing-vertex-relationship graph (CVR graph). We propose a novel structure, by which we offer an effective and easy operative method for constructing the basis functions of the biquadratic spline space. The mapping we construct is an isomorphism. The basis functions of the biquadratic spline space hold the properties such as linearly independent, completeness and the property of partition of unity, which are the same with the properties for the basis functions of piecewise constant space over the CVR graph. To demonstrate that the new basis functions are efficient, we apply the basis functions to fit some open surfaces.