Disk B-spline on 𝕊2: A Skeleton-based Region Representation on 2-Sphere

Abstract

Due to the widespread applications of 2-dimensional spherical designs, there has been an increasing requirement of modeling on the 𝕊² manifold in recent years. Due to the non-Euclidean nature of the sphere, it has some challenges to find a method to represent 2D regions on 𝕊² manifold. In this paper, a skeleton-based representation method of regions on 𝕊², disk B-spline(DBSC) on 𝕊² is proposed. Firstly, we give the definition and basic algorithms of DBSC on 𝕊². Then we provide the calculation method of DBSC on 𝕊², which includes calculating the boundary points, internal points and their corresponding derivatives. Based on that, we give some modeling methods of DBSC on 𝕊², including approximation, deformation. In the end, some stunning application examples of DBSC on 𝕊² are shown. This work lays a theoretical foundation for further applications of DBSC on 𝕊².