Fast spherical mapping for genus-0 meshes

Parameterizing a genus-0 mesh onto a sphere means assigning a 3D position on the unit sphere to each of the mesh vertices, such that the spherical mapping induced by the mesh connectivity is not too distorted and does not have overlapping areas. Satisfying the non-overlapping requirement sometimes is the most difficult and critical component of many spherical parametrization methods. In this paper we propose a fast spherical mapping approach which can map any closed genus-0 mesh onto a unit sphere without overlapping any part of the given mesh. This new approach does not try to preserve angles or edge lengths of the given mesh in the mapping process, however, test cases show it can obtain meaningful results. The mapping process does not require setting up any linear systems, nor any expensive matrix computation, but is simply done by iteratively moving vertices of the given mesh locally until a desired spherical mapping is reached. Therefore the new spherical mapping approach is fast and consequently can be used for meshes with large number of vertices. Moreover, the iterative process is guaranteed to be convergent. Our approach can be used for texture mapping, remeshing, 3D morphing, and can be used as input for other more rigorous and expensive spherical parametrization methods to achieve more accurate parametrization results. Some test results obtained using this method are included and they demonstrate that the new approach can achieve spherical mapping results without any overlapping.