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“Dihedral Angle-based Maps of Tetrahedral Meshes”

Gilles-Philippe Paillé, Nicolas Ray, Pierre Poulin, Alla Sheffer and Bruno Lévy

ACM Transactions on Graphics (SIGGRAPH 2015 Conf. Proc.), 2015

Abstract: We present a geometric representation of a tetrahedral mesh that is solely based on dihedral angles. 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 R3. 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.

## BibTex reference

@ARTICLE{PRPSL15,
AUTHOR = "Gilles-Philippe Paillé and Nicolas Ray and Pierre Poulin and A
lla Sheffer and Bruno Lévy",
TITLE = "Dihedral Angle-based Maps of Tetrahedral Meshes",
JOURNAL = "ACM Transactions on Graphics (SIGGRAPH 2015 Conf. Proc.)",
VOLUME = "34",
NUMBER = "4",
YEAR = "2015",
}

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