Fast C 2 Interpolating Subdivision Surfaces using Iterative Inversion of Stationary Subdivision Rules

This paper presents a simple iterative algorithm for computing an initial mesh which interpolates the vertices of a target base-mesh in the limit under Catmull-Clark subdivi- sion rules. It uses only local 1-neighborhood information at each vertex of the target mesh, and runs in O(nm) time where n is the number of vertices in the base mesh and m is the number of iterations. Since m typically small gives interpolation to within acceptable tolerances, the algorithm is effectively O(n), requiring only constant work for each base vertex to achieve an almost-everywhere C 2 interpolating surface. This algorithm is effective for interpolation of closed, two sided meshes and assumes non-singularity of the underlying global splitting matrix. While the argument presented is for Catmull-Clark subdivision, it may be trivially extended to include Loop subdivision of tri-meshes as well. This method may be modified in a quick-and-dirty fashion to interpolate boundary normals as well as positions at the desired mesh points; unlike more sophisticated methods, however, no smoothing is done—the sur- face will thus have "ripples", fine-scale surface perturbations not implied by a coarse medial sampling.