Abstract Part-Based Representation and Editing

The idea that a complex object can be decomposed into simpler parts is fundamental to 3D design, so it is clearly desirable that digital representations of 3D shapes incorporate this part information. While solid modeling techniques based on set-theoretic volumet-ric composition intrinsically support hierarchical part-based shape descriptions, organic objects such as a human vertebra are more efficiently represented by surface modeling techniques. And although a human observer will easily identify part decompositions in surface models, the homogenous graphs of connected points and edges used in surface representations do not readily support explicit part decompositions. In this thesis, I will develop a part-based representation for 3D surface models. In abstract mathematics, a surface part can be represented as a deformation of a Riemannian manifold. To create a practical implementation, it is necessary to define representations of the 3D part shape and the region on the target surface where the part is to be placed. To represent the part region I will develop the Discrete Exponential Map (DEM), an algorithm which approximates the intrinsic normal coordinates on manifolds. To support arbitrary part shapes I will develop the COILS surface deformation, a robust geometri