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 volumetric 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 defne 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 geometric differential representation of point-sampled surfaces. Based on this part definition, I will then propose the Surface Tree, which makes possible the representation of complex shapes via a procedural, hierarchical composition of surface parts, analogous to the trees used in solid modeling.

A major theme throughout the thesis is that part-based approaches have the potential to make surface design interfaces significantly more efficient and expressive. To explore this question and demonstrate the utility of my technical contributions, I present three novel modeling tools: an interactive texture design interface, a drag-and-drop mesh composition tool, and a sketch-based Surface Tree modeling environment. In addition to comparative algorithmic evaluations, and a consideration of representational capabilities, I have evaluated this body of work by publicly distributing my modeling tools. I will close the thesis with a discussion of the extensive feedback provided by users of my drag-and-drop mesh composition tool, called meshmixer. This feedback suggests that part-based approaches have signicant benefits for surface modeling.

All the software described in my thesis has been released in some form or another:

Discrete Exponential Map (Chapter 4) - Demo w/ C++ source code

Decal Texturing Tool (Chapter 4) - implemented in ShapeShop

Drag-and-Drop Mesh Composition Tool (Chapter 5) - demo w/ source is available. Also implemented in Autodesk meshmixer

Surface Tree modeling tool (Chapter 6) - implemented in ShapeShop
(Note: I have a standalone Surface Tree implementation that I plan on releasing code for once I find time to clean it up. If you want to hack on it, Contact me)

The full sources to ShapeShop and meshmixer are not generally available. However, if you have an idea for a project that could be implemented inside one of these tools, Contact me and we can probably work something out.