C<scp>ube</scp>C<scp>over</scp>– Parameterization of 3D Volumes

TLDR

Quad‑based 2D surface parameterization methods have succeeded, but effective algorithms for 3D volumes with hexahedral elements remain lacking, and Cube C over extends the Quad C over approach from surfaces to volumes. Cube C over is a first approach for generating a hexahedral tessellation of a given volume with boundary‑aligned cubes guided by a frame field. It takes a tetrahedral volume mesh, designs a manually input frame field to guide interior and boundary layout, and computes a parameterization and hexahedral mesh aligned with that field. The paper presents theoretical results for 3D hexahedral parameterizations and analyzes the topological properties of the corresponding function space.

Abstract

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.