|Title:||Topics in Shape Optimization and Exploration
|Abstract:||Shapes are the primary objects of interest in the field of geometry processing. Shape optimization is the task of improving a defined set of features of a shape, while satisfying some constraints. Shape exploration is concerned with finding variations to a shape or deforming it, using a user-friendly interface. In the first part of this work, we discuss two classical problems in geometry processing. The first problem is surface reconstruction from point clouds, where a surface is fit to a set of points, sampled from a real world object. We provide a generalization of the well-known Radial Basis Function approach and show that it outperforms other surface reconstruction algorithms. The second problem is 2D shape deformation, where we derive a new type of barycentric coordinates that have higher order interpolation abilities. The second part of this work is oriented toward the field known as architectural geometry. We discuss Polyhedral Meshes (PMs), i.e. meshes with planar faces, which are commonly used in architectural design. We provide a specialized system that enables real time editing of PMs, but nonetheless general enough to handle many types of constraints, which can be handled in interactive times. In addition, we analyze the manifold of PMs, and show that it can be decomposed into a set of maximal linear subspaces. Based on this decomposition, we provide new tools for exploring this manifold, which are more powerful than the state-of-the-art.|
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