A general construction of spherical barycentric coordinates and applications

Abstract

We provide a general method for defining and efficiently computing barycentric coordinates 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.