# Shape from Metric

Albert Chern, Felix Knöppel, Franz Pedit, Ulrich Pinkall, Peter Schröder

### Description

The \textit{isometric immersion} problem for orientable surface is interesting for smooth surfaces [1], as well as for triangle meshes [2] endowed with only a metric: given the combinatorics of the mesh together with edge lengths, approximate an isometric immersion into $\mathbb{R}^3$ .

To address this challenge a discrete theory for surface immersions into $\mathbb{R}^3$ has been developed. 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.

A full implementation is provided on the project site http://page.math.tu-berlin.de/~chern/projects/ShapeFromMetric/ .

### References

#### Dr. Albert Chern   +

University: TU Berlin
E-Mail: chern[at]math.tu-berlin.de

#### Dr. Felix Knöppel   +

Projects: C07
University: TU Berlin
E-Mail: knoeppel[at]math.tu-berlin.de

#### Prof. Dr. Ulrich Pinkall   +

Projects: A05, C07
University: TU Berlin, Institut für Mathematik, MA 822
Address: Straße des 17. Juni 136, 10623 Berlin, GERMANY
Tel: +49 30 31424607
E-Mail: pinkall[at]math.tu-berlin.de
Website: http://page.math.tu-berlin.de/~pinkall/