Volume and shape preservation via moving frame manipulation

This article introduces a method for mesh editing that is aimed at preserving shape and volume. We present two new developments: The first is a minimization of a functional expressing a geometric distance measure between two isometric surfaces. The second is a local volume analysis linking the volume of an object to its surface curvature. Our method is based upon the moving frames representation of meshes. Applying a rotation field to the moving frames defines an isometry. Given rotational constraints, the mesh is deformed by an optimal isometry defined by minimizing the distance measure between original and deformed meshes. The resulting isometry nicely preserves the surface details, but when large rotations are applied, the volumetric behavior of the model may be unsatisfactory. Using the local volume analysis, we define a scalar field by which we scale the moving frames. Scaled and rotated moving frames restore volumetric properties of the original mesh, while properly maintaining the surface details. Our results show that even extreme deformations can be applied to meshes, with only minimal distortion of surface details and object volume.