Global Surface Parameterization by Smooth Facet Selection

We present a global surface parameterization for smooth, star-like domains with asymptotic singu-larities (edges and corners) whose “sharpness ” can be tuned. The basic idea is a novel utilization of the (near-)maximum norm. For a given parameterized direction, by choosing the largest inverse distance, it “selects ” out of a set of limiting facets the one which is closest. Thus each facet’s orienta-tion and location can be individually controlled. Assembly of the final surface happens automatically and C-smoothly over the entire parameter domain, i.e., the unit circle (2D) or unit sphere (3D). The resulting surface is directly amenable to fast Fourier transforms and related methods and should also be useful for many high-order numerical solvers. We demonstrate how to construct general convex polygons and polyhedra and point out a few possibilities for concave, star-like domains. Keywords: 1