Rectangular Surface Parameterization

This paper describes a method for computing surface parameterizations that map infinitesimal axis-aligned squares in the plane to infinitesimal rectangles on the surface. Such rectangular parameterizations are needed for a broad range of tasks, from physical simulation to geometric modeling to computational fabrication. Our main contribution is a novel strategy for constructing frame fields that are perfectly orthogonal and exactly integrable, in the limit of mesh refinement. In contrast to past strategies for achieving integrability, we obtain maps that are less distorted and better preserve target field directions. The method supports user-defined distortion measures, sharp feature alignment, prescribed or automatic cone singularities, and direct control over boundary behavior (e.g., sizing or aspect ratio). By quantizing and contouring these maps we obtain high-quality anisotropic quad meshes, even without element-based optimization. Empirically, we outperform state-of-the-art research and commercial mesh generation algorithms in terms of element quality, accuracy, and asymptotic convergence rate in end-to-end simulation tasks, are competitive with the widely-used ZBrush package for automatic retopology, and provide Chebyshev nets of superior quality to methods specifically tailored to digital fabrication.