Dihedral angle-based maps of tetrahedral meshes

Gilles-Philippe Paillé (1) , Nicolas Ray (2) , Pierre Poulin (3) , Alla Sheffer (4) , Bruno Lévy (2)

1 DIRO - Département d'Informatique et de Recherche Opérationnelle [Montreal]
2 ALICE - Geometry and Lighting
3 CRPP - Centre de Recherche Paul Pascal
4 UBC-Computer Science - Computer Science Department

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 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.

Mots clés: Volume parameterization; interpolation; compression; optimization

Domaines: Géométrie algorithmique [cs.CG]

Gilles-Philippe Paillé, Nicolas Ray, Pierre Poulin, Alla Sheffer, Bruno Lévy. Dihedral angle-based maps of tetrahedral meshes. SIGGRAPH 2015 : 42nd International Conference and Exhibition on Computer Graphics and Interactive Techniques, Aug 2015, Los Angeles United States. ⟨10.1145/2766900⟩. ⟨hal-01245593⟩