Dupin Meshing: A Parameterization Approach to Planar Hex-Dominant Meshing

Planar hexagonal-dominant (PHex) meshes are an important class of meshes with minimal vertex-degree. They are highly useful in the rationalization of freeform architectural surfaces, for construction with flat steel, glass, or wooden panels of equal thickness. A PHex mesh must contain both convex and concave faces of varying anisotropic shapes due to the planarity constraint. Therefore, while parameterization-based approaches have been successfully used for planar-quad meshing, applying such approaches to PHex meshing has not been attempted so far. In this paper, we show how to bridge this gap, effectively allowing us to leverage the flexibility of quad-remeshing methods (e.g. field alignment, singularities) to PHex remeshing. We have two main observations. First, the anisotropy can be handled by isotropically remeshing a modified geometry, which we denote by curvature shape, and then pulling back the result to the original surface. Second, various non-convex face shapes can be generated robustly by locally modifying the grid texture used for discretizing the parametrization. Together, these reductions yield a simple and effective method for PHex remeshing of triangular surfaces, which is additionally robust, and applicable to a variety of models. We compare our method to recent state-ofthe-art methods for PHex meshing and demonstrate the advantages of our approach.