Volumetric Quasi-conformal Mappings - Quasi-conformal Mappings for Volume Deformation with Applications to Geometric Modeling

Due to intrinsic differences between surfaces and higher dimensional objects, some important results regarding surfaces can not be extended to volumetric domains. Most significantly, there exist no conformal volumetric maps apart from Mobius transformations. Although it is sometime stated explicitly, it is often overlooked that existing methods of volume parameterization produce only quasi-conformal maps, which may be “far from conformality”. We therefore introduce methods for assessing the extent of the local and global volumetric deformation by means of the amount of conformal distortion produced. To this end we first illustrate basic three-dimensional quasi-conformal deformations that are produced by parameterization techniques, and highlight theoretical issues associated with spatial quasi-conformal mappings, and the relation that exists between the geometry of the domain and conformal distortion.