We introduce hexagonal global parameterization, a new type of surface parameterization in which parameter lines respect sixfold rotational symmetries (6-RoSy). Such parameterizations enable the tiling of surfaces with nearly regular hexagonal or triangular patterns, and can be used for triangular remeshing. Our framework to construct a hexagonal parameterization, referred to as HEXCOVER, extends the QUADCOVER algorithm and formulates necessary conditions for hexagonal parameterization. We also provide an algorithm to automatically generate a 6-RoSy field that respects directional and singularity features in the surface. We demonstrate the usefulness of our geometry-aware global parameterization with applications such as surface tiling with nearly regular textures and geometry patterns, as well as triangular and hexagonal remeshing.

Abstract Despite the success of quad‐based 2D surface parameterization methods, effective parameterization algorithms for 3D volumes with cubes, i.e. hexahedral elements, are still missing. C ube C over is a first approach for generating a hexahedral tessellation of a given volume with boundary aligned cubes which are guided by a frame field. The input of C ube C over is a tetrahedral volume mesh. First, a frame field is designed with manual input from the designer. It guides the interior and boundary layout of the parameterization. Then, the parameterization and the hexahedral mesh are computed so as to align with the given frame field. C ube C over has similarities to the Q uad C over algorithm and extends it from 2D surfaces to 3D volumes. The paper also provides theoretical results for 3D hexahedral parameterizations and analyses topological properties of the appropriate function space.

This sketch introduces hexagonal global parameterization, a new type of periodic global parameterization that is ideal for tiling surfaces with patterns of six-fold rotational symmetries, i.e., 6-RoSy's [Palacios and Zhang 2007]. Being one of the two most fundamental rotational symmetries that are compatible with translational symmetries in the plane, 6-RoSy's appear in many places in nature, such as honeycombs, insect eyes, corals and crystals, as well as man-made objects such as Islamic patterns [Kaplan and Salesin 2004] and tri-axial weaving [Akleman et al. 2009]. Such symmetries can also provide optimal circle packing, which naturally have applications in architectural design [Schiftner et al. 2009]. A hexagonal global parameterization facilitates all of these applications. See Figure 1 for some examples. Furthermore, parameter lines in a hexagonal global parameterization intersect at an angle of π/3, which enables triangular remeshing of a mesh surface with close to ideal aspect ratios in the triangles. The dual mesh of such a triangulation provides a hexagon-dominant tiling of the surface.

Abstract We introduce an algorithm for the automatic computation of global parameterizations on arbitrary simplicial 2‐manifolds, whose parameter lines are guided by a given frame field, for example, by principal curvature frames. The parameter lines are globally continuous and allow a remeshing of the surface into quadrilaterals. The algorithm converts a given frame field into a single vector field on a branched covering of the 2‐manifold and generates an integrable vector field by a Hodge decomposition on the covering space. Except for an optional smoothing and alignment of the initial frame field, the algorithm is fully automatic and generates high quality quadrilateral meshes.