Dihedral Rigidity and Deformation

TL;DR

This paper explores how dihedral angles can uniquely determine a triangle mesh's shape and demonstrates a method for creating smooth shape morphs by interpolating these angles.

Contribution

It establishes the local one-to-one correspondence between dihedral angles and mesh embeddings and introduces a heuristic for shape interpolation in dihedral space.

Findings

Dihedral angles can locally define mesh embeddings uniquely.

Interpolations in dihedral space produce smooth shape morphs.

The method offers intuitive shape transformations.

Abstract

We consider defining the embedding of a triangle mesh into $R^3$, up to translation, rotation, and scale, by its vector of dihedral angles. Theoretically, we show that locally, almost everywhere, the map from realizable vectors of dihedrals to mesh embeddings is one-to-one. We experiment with a heuristic method for mapping straight-line interpolations in dihedral space to interpolations between mesh embeddings and produce smooth and intuitively appealing morphs between three-dimensional shapes.