A general construction of spherical barycentric coordinates and applications

We provide a general method for defining and efficiently computing barycentric coordi-nates with respect to polygons on the unit sphere. More precisely, we develop a novel explicit construction which allows us to compute the spherical barycentric coordinates from their 2D-Euclidean counterparts. In particular, we give two interesting families of spherical coordinates, one is defined for convex and non-convex spherical polygons. An interesting consequence is the possibility to construct new 3D barycentric coordinates for arbitrary polygonal meshes. Furthermore, we present an alternative construction for spherical barycentric coordinates with help of 3D barycentric coordinates for closed triangular meshes. This construction is extended to arbitrary dimensions. We show that our spherical and 3D coordinates are widely applicable to many domains. We give several examples related to spherical blending, space deformations and shape morphing in 3D. (c) 2022 Elsevier B.V. All rights reserved.

Automated summary

We propose a general method for defining and efficiently computing barycentric coordinates with respect to polygons on the unit sphere. We develop a novel explicit construction to compute the spherical barycentric coordinates from their 2D-Euclidean counterparts and provide families of spherical coordinates for convex and non-convex spherical polygons. We also present an alternative construction for spherical barycentric coordinates using 3D barycentric coordinates for closed triangular meshes, which can be extended to arbitrary dimensions. Our spherical and 3D coordinates have wide applicability in various domains, demonstrated through examples in spherical blending, space deformations, and shape morphing in 3D.