Tangent-ball techniques for shape processing

Shape processing defines a set of theoretical and algorithmic tools for creating, measuring and modifying digital representations of shapes.  Such tools are of paramount importance to many disciplines of computer graphics, including modeling, animation, visualization, and image processing.  Many applications of shape processing can be found in the entertainment and medical industries. In an attempt to improve upon many previous shape processing techniques, the present thesis explores the theoretical and algorithmic aspects of a difference measure, which involves fitting a ball (disk in 2D and sphere in 3D) so that it has at least one tangential contact with each shape and the ball interior is disjoint from both shapes. We propose a set of ball-based operators and discuss their properties, implementations, and applications.  We divide the group of ball-based operations into unary and binary as follows: Unary operators include: * Identifying details (sharp, salient features, constrictions) * Smoothing shapes by removing such details, replacing them by fillets and roundings * Segmentation (recognition, abstract modelization via centerline and radius variation) of tubular structures Binary operators include: * Measuring the local discrepancy between two shapes * Computing the average of two shapes * Computing point-to-point correspondence between two shapes * Computing circular trajectories between corresponding points that meet both shapes at right angles * Using these trajectories to support smooth morphing (inbetweening) * Using a curve morph to construct surfaces that interpolate between contours on consecutive slices The technical contributions of this thesis focus on the implementation of these tangent-ball operators and their usefulness in applications of shape processing. We show specific applications in the areas of animation and computer-aided medical diagnosis.  These algorithms are simple to implement, mathematically elegant, and fast to execute.Ph.D.Committee Chair: Jarek Rossignac; Committee Member: Greg Slabaugh; Committee Member: Greg Turk; Committee Member: Karen Liu; Committee Member: Maryann Simmon

The topology and geometry of liquid crystals

Liquid crystals are materials that exhibit a number of fascinating properties, many of which have a geometric and topological flavour. Our understanding of liquid crystals often comes through the study of their topological defects, which has inspired new concepts of structural organization in soft matter. Topological and geometric methods have been fundamental to these developments. Today it is impossible to imagine any direction of the science of liquid crystals that does not actively use the concepts of topological defects: they play an essential role in fundamental theory and descriptions of such materials, and also underpin the promising new applications. Varieties of liquid crystal with additional geometric structure are at the forefront of new applications. These materials include the chiral nematics, or cholesterics, and also the more recently discovered twist-bend nematics. It is known that chirality especially brings enormous richness, allowing for a wealth of new metastable states and textures. The defects in these classes of material have more structure than those in standard nematics, and include not just defects in the director, but defects in other directions associated to the director which nonetheless have a fundamental structural importance; these are the familiar lambda lines of cholesterics, and the beta lines in twist-bend materials which I describe for the first time. Despite their importance, defects in cholesteric materials are still poorly understood. In this thesis I develop a theory of point and line defects in cholesterics using the mathematics of contact topology. I classify the structure of point defects by using singularity theory and contact topology; the classification shows a very good correspondence with experimental observations. Hedgehog point defects, ubiquitous in standard nematics, are energetically disfavoured in cholesterics due to being incompatibile with a single handedness. The same constraint applies to the boundary of a droplet with normal anchoring, which results in topologically-protected regions of reversed handedness in the boundary region. These ‘twist solitons’ are a novel type of topological defect in cholesterics, identified and studied here for the first time. This theory is applied to recent experiments to explain the stability of the novel structures observed in spherical cholesteric droplets. Additional textures with complex layered structures are examined from the perspective of contact topology. Convex surface theory aids visualisation of layered structures as well as helping to describe their properties. I give an overview of the structures that may occur based on their layer topology, achieving a good correspondence with experiment. Disclinations in cholesterics are studied using contact topology, and a full classification up to homotopy is obtained. The dichotomy between tight and overtwisted structures manifests itself in an interesting way for disclinations: the tight disclinations are exactly those not attached to positive strength lambda lines. While the overtwisted lines admit no new invariants, for these tight lines I identify a novel topological invariant, which has the form of a self-linking number. Orientable singular lines can be removed in a standard nematic, but the classification shows obstructions to doing this in a cholesteric which I use to explain certain experimentally-observed textures. This observation also leads to an experimentally accessible method for generating metastable twist solitons in the bulk, as well as suggesting a novel method for generating Hopf solitons. The topological invariants of nematics can be related to the zeros of a vector field orthogonal to the director. Using this approach, I study Hopf solitons in cholesteric droplets using the lambda lines, a novel perspective. The bend distortion of the director is a vector field that is always orthogonal to it; its zeros, the beta lines, have fundamental importance in the twist-bend nematic phase. I produce the first topological and geometric study of this phase, identifying various textures and defects, including Skyrmions, screw dislocations, and focal conics, by the structure of their beta lines. Hopf solitons in twist-bend materials appear not be stable; fundamental results in contact topological give insight into their process of removal, as well demonstrating that they are replaced with a twist soliton. Finally, I develop a geometric theory of directors using Cartan’s method of moving frames. As well as giving new insight into the director distortions and the relationships between them, this allows us to study the problem of reconstructing a director from its gradients, which has previously been solved in two dimensions but not three dimensions. This approach demonstrates the connection between directors and Lie theory, and suggests a description of directors in terms of their local symmetry groups