Thesis (Ph. D.)--University of Washington, 2002. / Vita. Includes bibliographical references (leaves 87-88).

Thesis (M.S.)--University of Missouri-Columbia, 2007. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed Oct. 26, 2007). Vita. Includes bibliographical references.

Here we investigate a variety of ways to represent polytopes and related objects. We define a class of posets, which includes all abstract polytopes, giving a unique representative among posets having a particular labeled flag graph and characterize the labeled flag graphs of abstract polytopes. We show that determining the realizability of an abstract polytope is equivalent to solving a low rank matrix completion problem. For any given polytope, we provide a new construction for the known result that there is a combinatorial polytope with a specified ridge that is always projectively equivalent to the given polytope, and we show how this makes a naturally arising subclass of intractable problems tractable. We give necessary and sufficient conditions for realizing a polytope's interval poset, which is the polytopal analog of a poset's Hasse diagram. We then provide a counter example to the general realizablity of a polytope's interval poset. / Mathematics

Thesis (Ph. D.)--Rutgers University, 2010. / "Graduate Program in Computer Science." Includes bibliographical references (p. 62-64).

Toward support of the use of geometry in advanced simulation, a freeorm deformation (FFD) tool was designed, developed, and tested using object oriented (OO) techniques. The motivation for creating this FFD tool in-house was to provide engineers and researchers a cost efficient, quick, and easy way to computationally manipulate models without having to start from scratch while readily seeing the resulting geometry. The FFD tool was built using the OO scripting language, Python, the OO GUI toolkit, Qt, and the graphics toolkit, OpenGL. The tool produced robust and intuitive results for two-dimensional shapes especially when multiple point manipulation was utilized. The use of ?grouping? control points also provided the user the ability to maintain certain desired shape features such as straight lines and corners. This in-house FFD tool could be useful to engineers due to the ability to customize source code.

Recent innovations in design and construction of architectural buildings has led us to revisit the metrics for discretizing smooth freeform shapes in context with both aesthetics and fabrication. Inspired by the examples of the British Museum Court Roof in Britain and the Beijing Aquatic Centre in China, we propose solutions for generating aesthetic as well as economically viable solutions for tessellating smooth, freeform shapes. For the purpose of generating an aesthetic tessellation, we propose a simple linearized strain based metric to minimize dissimilarity amongst triangles in a local neighborhood. We do so by defining an error function that measures deformation required to map a pair of triangles onto each other. We minimize the error using a global non-linear optimization based framework. We also reduce the complexity associated with prefabricating triangulated panels for a given shape. To do so, we propose a global optimization based framework to approximate any given shape using significantly reduced numbers of unique triangles. By doing so, we leverage the economies of scale as well as simplify the process of physical placement of panels by manual labor.

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The computation of curvature quantities over discrete geometry is often required when processing geometry composed of meshes. Curvature information is often important for the purpose of shape analysis, feature recognition and geometry segmentation. In this paper we present a method for accurate estimation of curvature on discrete geometry especially those composed of meshes. We utilise a method based on fitting a continuous surface arising from the solution of the Biharmonic equation subject to suitable boundary conditions over a 1-ring neighbourhood of the mesh geometry model. This enables us to accurately determine the curvature distribution of the local area. We show how the curvature can be computed efficiently by means of utilising an analytic solution representation of the chosen Biharmonic equation. In order to demonstrate the method we present a series of examples whereby we show how the curvature can be efficiently computed over complex geometry which are represented discretely by means of mesh models.

The results of this thesis concern folding of one-dimensional objects in two dimensions: planar linkages. More precisely, a planar linkage consists of a collection of rigid bars (line segments) connected at their endpoints. Foldings of such a linkage must preserve the connections at endpoints, preserve the bar lengths, and (in our context) prevent bars from crossing. The main result of this thesis is that a planar linkage forming a collection of polygonal arcs and cycles can be folded so that all outermost arcs (not enclosed by other cycles) become straight and all outermost cycles become convex. A complementary result of this thesis is that once a cycle becomes convex, it can be folded into any other convex cycle with the same counterclockwise sequence of bar lengths. Together, these results show that the configuration space of all possible foldings of a planar arc or cycle linkage is connected. These results fall into the broader context of folding and unfolding <I>k</I>-dimensional objects in <i>n</i>-dimensional space, <I>k</I> less than or equal to <I>n</I>. Another contribution of this thesis is a survey of research in this field. The survey revolves around three principal aspects that have received extensive study: linkages in arbitrary dimensions (folding one-dimensional objects in two or more dimensions, including protein folding), paper folding (normally, folding two-dimensional objects in three dimensions), and folding and unfolding polyhedra (two-dimensional objects embedded in three-dimensional space).