In this thesis, a novel adaptive method for tracking moving surfaces is presented. Our methods are based on a framework for surface propagation, called face offsetting, that moves the mesh faces independently, and reconstructs vertex locations using an eigendecomposition of an error metric together with a viscosity adjustment. Our method prevents local self-intersections, which was a significant roadblock in moving surface meshes. This new framework offers a compelling alternative to level set methods for geometric processing of surface meshes because of its convenience, higher efficiency, and volume conservation.Utilizing the face-offsetting framework, we also develop a new technique for surface smoothing that preserves volume locally (instead of globally) via keeping track of a height function. Volume preservation is critical in many applications, but typical preserving methods rely on a global perturbation to the surface and hence suffer from undesirable side effects.The level sets method propagates a surface represented as an isosurface of a volume of scalar values that automatically accommodates the self-intersection and topology changes that can occur during propagation. The level set method propagates the isosurface in its normal direction according to a user-defined speed function evaluated over it, by deriving and integrating a corresponding time derivative of the voxel values.We apply the power of the level set method to the propagation of an implicit surface represented not as the interpolation of voxel values but more conventionally through the conglomeration of simpler primitive shapes.The proposed strategy retains topological benefits of the implicit representation of evolving surfaces while avoiding the drawbacks of a fixed resolution voxel array. This propagation of course occurs within the limits of the implicit's parametrization, and our method creates a least-squares optimal fit of the implicit to the shape specified by the geometric flow.Morse theory reveals the topological structure of a shape based on the critical points of a real function over the shape. This thesis solves a relaxed form of Laplace's equation to find a "fair" Morse function with a user-controlled number and configuration of critical points.
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