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 \(\Bbb R^3\). To address this challenge we develop a discrete theory for surface immersions into \(\Bbb R^3\). 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.
This work was supported in part by the DFG Collaborative Research Center TRR 109 "Discretization in Geometry and Dynamics." Additional support was provided by SideFX software. The Bunny model is courtesy Stanford Computer Graphics laboratory and the Duck model courtesy Keenan Crane.