Abstract: The traditional approach for parametrizing a surface involves cutting it into charts and mapping these piecewise onto a planar domain. We introduce a robust technique for directly parametrizing a genus-zero surface onto a spherical domain. A key ingredient for making such a parametrization practical is the minimization of a stretch-based measure, to reduce scaledistortion and thereby prevent undersampling. Our second contribution is a scheme for sampling the spherical domain using uniformly subdivided polyhedral domains, namely the tetrahedron, octahedron, and cube. We show that these particular semiregular samplings can be conveniently represented as completely regular 2D grids, i.e. geometry images. Moreover, these images have simple boundary extension rules that aid many processing operations. Applications include geometry remeshing, level-ofdetail, morphing, compression, and smooth surface subdivision.