This thesis covers work on two topics: unfolding polyhedra into the plane and reconstructing polyhedra from partial information. For each topic, we describe previous work in the area and present an array of new research and results. Our work on unfolding is motivated by the problem of characterizing precisely when overlaps will occur when a polyhedron is cut along edges and unfolded. By contrast to previous work, we begin by classifying overlaps according to a notion of locality. This classification enables us to focus upon particular types of overlaps, and use the results to construct examples of polyhedra with interesting unfolding properties. The research on unfolding is split into convex and non-convex cases. In the non-convex case, we construct a polyhedron for which every edge unfolding has an overlap, with fewer faces than all previously known examples. We also construct a non-convex polyhedron for which every edge unfolding has a particularly trivial type of overlap. In the convex case, we construct a series of example polyhedra for which every unfolding of various types has an overlap. These examples disprove some existing conjectures regarding algorithms to unfold convex polyhedra without overlaps. The work on reconstruction is centered around analyzing the computational complexity of a number of reconstruction questions. We consider two classes of reconstruction problems. The first problem is as follows: given a collection of edges in space, determine whether they can be rearranged <em>by translation only</em> to form a polygon or polyhedron. We consider variants of this problem by introducing restrictions like convexity, orthogonality, and non-degeneracy. All of these problems are NP-complete, though some are proved to be only weakly NP-complete. We then consider a second, more classical problem: given a collection of edges in space, determine whether they can be rearranged by <em>translation and/or rotation</em> to form a polygon or polyhedron. This problem is NP-complete for orthogonal polygons, but polynomial algorithms exist for non-orthogonal polygons. For polyhedra, it is shown that if degeneracies are allowed then the problem is NP-hard, but the complexity is still unknown for non-degenerate polyhedra.

This thesis covers work on two topics: unfolding polyhedra into the plane and reconstructing polyhedra from partial information. For each topic, we describe previous work in the area and present an array of new research and results. Our work on unfolding is motivated by the problem of characterizing precisely when overlaps will occur when a polyhedron is cut along edges and unfolded. By contrast to previous work, we begin by classifying overlaps according to a notion of locality. This classification enables us to focus upon particular types of overlaps, and use the results to construct examples of polyhedra with interesting unfolding properties. The research on unfolding is split into convex and non-convex cases. In the non-convex case, we construct a polyhedron for which every edge unfolding has an overlap, with fewer faces than all previously known examples. We also construct a non-convex polyhedron for which every edge unfolding has a particularly trivial type of overlap. In the convex case, we construct a series of example polyhedra for which every unfolding of various types has an overlap. These examples disprove some existing conjectures regarding algorithms to unfold convex polyhedra without overlaps. The work on reconstruction is centered around analyzing the computational complexity of a number of reconstruction questions. We consider two classes of reconstruction problems. The first problem is as follows: given a collection of edges in space, determine whether they can be rearranged <em>by translation only</em> to form a polygon or polyhedron. We consider variants of this problem by introducing restrictions like convexity, orthogonality, and non-degeneracy. All of these problems are NP-complete, though some are proved to be only weakly NP-complete. We then consider a second, more classical problem: given a collection of edges in space, determine whether they can be rearranged by <em>translation and/or rotation</em> to form a polygon or polyhedron. This problem is NP-complete for orthogonal polygons, but polynomial algorithms exist for non-orthogonal polygons. For polyhedra, it is shown that if degeneracies are allowed then the problem is NP-hard, but the complexity is still unknown for non-degenerate polyhedra.

Thesis (Ph. D.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; contains ix, 118 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 115-118).

Algorithms to generate walks (chains of unit-length, freely-jointed segments) and polygons (closed walks) in spherical confinements have been developed in the last few years. These algorithms generate polygons inside spherical confinement based on their mathematically derived probability distributions. The generated polygons do not occupy any volume { although that would be useful for some applications. This thesis investigates how to generate walks and polygons which occupy some volume in spherical confinement. More specifically, in this thesis, existing methods described in the literature have been studied and implemented to generate walks and polygons in confinement. Additionally, these methods were adapted to design, develop, and implement an algorithm which generates walks and polygons in confinement with thick segments, that is, segments which occupy volume. Data is collected by generating walks and polygons of different lengths with and without thickness inside the spherical confinements of various radii to compare walks and polygons with thickness with those generated without thickness. The analysis of the collected data shows that a. the newly developed algorithm indeed generates polygons which are thicker than those generated with the volumeless algorithm; and b. the newly developed algorithm generates polygons which are different from the polygons generated by the volumeless algorithm. The analysis also includes an assessment of the computational cost of generating thick polygons.

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The main topic of this thesis is the tropicalizations of Severi varieties, which we call tropical Severi varieties. Severi varieties are classical objects in algebraic geometry. They are parameter spaces of plane nodal curves. On the other hand, tropicalization is an operation defined in tropical geometry, which turns subvarieties of an algebraic torus into certain polyhedral objects in real vector spaces. By studying the tropicalizations, it may be possible to transform algebro-geometric problems into purely combinatorial ones. Thus, it is a natural question, “what are tropical Severi varieties?” In this thesis, we give a partial answer to this question: we obtain a description of tropical Severi varieties in terms of regular subdivisions of polygons. Given a regular subdivision of a convex lattice polygon, we construct an explicit parameter space of plane curves. This parameter space is a much simpler object than the corresponding Severi variety and it is closely related to a flat degeneration of the Severi variety, which in turn describes the tropical Severi variety. We present two applications. First, we understand G.Mikhalkin’s correspondence theorem for the degrees of Severi varieties in terms of tropical intersection theory. In particular, this provides a proof of the independence of point-configurations in the enumeration of tropical nodal curves. The second application is about Secondary fans. Secondary fans are purely combinatorial objects which parameterize all the regular subdivisions of polygons. We provide a relation between tropical Severi varieties and Secondary fans.

Thesis (M. Sc.)--Memorial University of Newfoundland, 1997. / Bibliography: leaves 74-75.

Thesis (M.C.S.)--Carleton University, 2001. / Includes bibliographical references (p. 75-81). Also available in electronic format on the Internet.

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