Shape from Metric (SIGGRAPH 2018)

This is the supplementary video for the paper "Shape from Metric," where we develop theory and an algorithm for the discrete isometric immersion problem: given a triangle mesh with only lengths of edges known, map the mesh to 3D isometrically without singularities.

Publication:
Albert Chern, Felix Knöppel, Ulrich Pinkall, Peter Schröder.
"Shape from Metric"
ACM Transactions on Graphics (TOG) Vol. 37 Issue 4, Article 63 (August 2018)

Preprint
http://multires.caltech.edu/pubs/Shap...

Project page (preprint & source code included):
http://page.math.tu-berlin.de/~chern/...

Music composed by Lucas King (   • Darth Revan Theme | Sad Piano Version  ), re-performed by Albert Chern.

Abstract:
We study the isometric immersion problem for orientable surface triangle meshes endowed with only a metric: given the combinatorics of the mesh together with edge lengths, approximate an isometric immersion into ℝ³. To address this challenge we develop a discrete theory for surface immersions into ℝ³. It precisely characterizes a discrete immersion, up to subdivision and small perturbations. In particular our discrete theory correctly represents the topology of the space of immersions, i.e., the regular homotopy classes which represent its connected components. Our approach relies on unit quaternions to represent triangle orientations and to encode, in their parallel transport, the topology of the immersion. In unison with this theory we develop a computational apparatus based on a variational principle. Minimizing a non-linear Dirichlet energy optimally finds extrinsic geometry for the given intrinsic geometry and ensures low metric approximation error.
We demonstrate our algorithm with a number of applications from mathematical visualization and art directed isometric shape deformation, which mimics the behavior of thin materials with high membrane stiffness.