### Fundamentals of spherical parameterization for 3D meshes

Craig Gotsman, Xianfeng Gu and Alla Sheffer

ACM Transaction on Graphics, Vol 22, num 3, pages 358-363, 2003

Parameterization of 3D mesh data is important for many graphics
applications, in particular for texture mapping, remeshing and
morphing. Closed manifold genus-0 meshes are topologically
equivalent to a sphere, hence this is the natural parameter domain
for them. Parameterizing a triangle mesh onto the sphere means
assigning a 3D position on the unit sphere to each of the mesh
vertices, such that the spherical triangles induced by the mesh
connectivity are not too distorted and do not overlap. Satisfying
the non-overlapping requirement is the most difficult and critical
component of this process. We describe a generalization of the
method of barycentric coordinates for planar parameterization
which solves the spherical parameterization problem, prove its
correctness by establishing a connection to spectral graph theory
and show how to compute these parameterizations.