Dihedral Angle-based Maps of Tetrahedral Meshes
Gilles-Philippe Paillé, Nicolas Ray, Pierre Poulin, Alla Sheffer, Bruno Lévy
SIGGRAPH / ACM Transactions on Graphics 2015

graphical abstract

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.

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Bibtex

@article{PRPSL15,
    author = {Gilles-Philippe Paill\'e and Nicolas Ray and Pierre Poulin and Alla Sheffer and Bruno L\'evy },
    title = {Dihedral Angle-based Maps of Tetrahedral Meshes},
    journal = {ACM Trans. Graph.},
    volume= {34},
    number= {4},
    year = {2015}
}