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Section: New Results

Dihedral Angle-Based Maps of Tetrahedral Meshes

Participants : Nicolas Ray, Bruno Lévy.

This work is a collaboration with Gilles-Philippe Paillé (visiting), Pierre Poulin (U. de Montréal) and Alla Sheffer (UBC).

Given a 2D triangulation, it is well known that it is reasonably easy to reconstruct the shape of all the triangles from the sole data of the angles at the triangle corners, provided that they satisfy some constraints. In this project, we studied how this idea can be generalized in the volumetric setting. In other words, we proposed a geometric representation of a tetrahedral mesh that is solely based on dihedral angles, and what are the constraints that these dihedral angles need to satisfy to make that possible. We first show that the shape of a tetrahedral mesh is completely defined by its dihedral angles. This proof leads to a set of angular constraints that must be satisfied for an immersion to exist in ${ℝ}^3$. This formulation lets us easily specify conditions to avoid inverted tetrahedra and multiply-covered vertices, thus leading to locally injective maps. We then present a constrained optimization method that modifies input angles when they do not satisfy constraints. Additionally, we develop a fast spectral reconstruction method to robustly recover positions from dihedral angles. We demonstrate the applicability of our representation with examples of volume parameterization, shape interpolation, mesh optimization, connectivity shapes, and mesh compression. This work has been published in Transactions on Graphics [17] .