Bending Invariant Representations for Surfaces

Isometric surfaces share the same geometric structure also known as the ‘first fundamental form’. For example, all possible bending of a given surface, that include all length preserving deformations without tearing or stretching the surface, are considered to be isometric. We present a method to construct a bending invariant canonical form for such surfaces. This invariant representation is an embedding of the intrinsic geodesic structure of the surface in a finite dimensional Euclidean space, in which geodesic distances are approximated by Euclidean ones. The canonical representation is constructed by first measuring the inter geodesic distances between points on the surfaces. Next, multi-dimensional scaling (MDS) techniques are applied to extract a finite dimensional flat space in which geodesic distances are represented as Euclidean ones. The geodesic distances are measured by the efficient ‘fast marching on triangulated domains’ numerical algorithm. Applying this transform to various objects with similar geodesic structures (similar first fundamental form) maps isometric objects into similar canonical forms. We show a simple surface classification method based on the bending invariant canonical form.