Ng Lai Sze. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 135-140). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Spherical Data Representation --- p.4 / Chapter 3 --- Rhombic Dodecahedron --- p.7 / Chapter 3.1 --- Introduction --- p.7 / Chapter 3.2 --- "Platonic Solids, Archimedean Solids and Dual Polyhedron" --- p.8 / Chapter 3.2.1 --- Platonic Solids --- p.8 / Chapter 3.2.2 --- Archimedean Solids --- p.10 / Chapter 3.2.3 --- Dual Polyhedron --- p.13 / Chapter 3.3 --- Rhombic Dodecahedron --- p.16 / Chapter 3.3.1 --- Basic Property of Rhombic Dodecahedron --- p.16 / Chapter 3.3.2 --- Construction of Rhombic Dodecahedron --- p.16 / Chapter 3.3.3 --- Advantages of Rhombic Dodecahedron --- p.16 / Chapter 3.4 --- Summary --- p.19 / Chapter 4 --- Subdivision Scheme --- p.21 / Chapter 4.1 --- Introduction --- p.21 / Chapter 4.2 --- Motivation --- p.22 / Chapter 4.3 --- Great Circle Subdivision --- p.22 / Chapter 4.3.1 --- Normal Space Analysis --- p.23 / Chapter 4.4 --- Small Circle Subdivision --- p.25 / Chapter 4.5 --- Skew Great Circle Subdivision --- p.27 / Chapter 4.6 --- Analysis --- p.28 / Chapter 4.6.1 --- Sampling Uniformity --- p.29 / Chapter 4.6.2 --- Area Uniformity --- p.32 / Chapter 4.6.3 --- Stretch Measurement --- p.35 / Chapter 4.6.4 --- Query Efficiency --- p.39 / Chapter 4.7 --- Summary --- p.40 / Chapter 5 --- Data Querying and Indexing --- p.42 / Chapter 5.1 --- Introduction --- p.42 / Chapter 5.2 --- Location of base polygon --- p.43 / Chapter 5.2.1 --- General Method --- p.43 / Chapter 5.2.2 --- Tailored Table Look Up Method --- p.45 / Chapter 5.3 --- Location of the subdivided area --- p.49 / Chapter 5.3.1 --- On Deriving the Indexing Equation --- p.50 / Chapter 5.4 --- Summary --- p.54 / Chapter 6 --- Environment Mapping --- p.56 / Chapter 6.1 --- Introduction --- p.56 / Chapter 6.2 --- Related Work --- p.57 / Chapter 6.3 --- Methodology --- p.58 / Chapter 6.4 --- Data Preparation --- p.59 / Chapter 6.4.1 --- Re-sampling of Data on Sphere --- p.60 / Chapter 6.4.2 --- Preparation of Texture --- p.65 / Chapter 6.5 --- Reflection and Refraction by environment mapping --- p.68 / Chapter 6.5.1 --- Location and Retrieval of Data --- p.68 / Chapter 6.5.2 --- Cg Implementation --- p.70 / Chapter 6.6 --- Experiments --- p.76 / Chapter 6.6.1 --- Experiment Setup --- p.76 / Chapter 6.6.2 --- Experiment Result and Analysis --- p.78 / Chapter 6.7 --- Summary --- p.89 / Chapter 7 --- Shadow Mapping --- p.92 / Chapter 7.1 --- Introduction --- p.92 / Chapter 7.2 --- Related Work --- p.93 / Chapter 7.3 --- Methodology --- p.95 / Chapter 7.4 --- Data Preparation --- p.97 / Chapter 7.5 --- Shadow Determination and Scene Illumination --- p.98 / Chapter 7.6 --- Experiments --- p.100 / Chapter 7.6.1 --- Experiment Setup --- p.100 / Chapter 7.6.2 --- Experiment Result and Analysis --- p.101 / Chapter 7.7 --- Summary --- p.107 / Chapter 8 --- Dynamic HDR Environment Sequences Sampling --- p.110 / Chapter 8.1 --- Introduction --- p.110 / Chapter 8.2 --- Related Work on HDR Distant Environment Map Sampling --- p.112 / Chapter 8.3 --- Static Sampling by Spherical Quad-Quad Tree --- p.114 / Chapter 8.3.1 --- Importance Metric --- p.117 / Chapter 8.4 --- Dynamic Sampling by Spherical Quad-Quad Tree --- p.121 / Chapter 8.5 --- Experiments --- p.125 / Chapter 8.5.1 --- Static Sampling --- p.125 / Chapter 8.5.2 --- Dynamic Sampling --- p.126 / Chapter 8.6 --- Summary --- p.132 / Chapter 9 --- Conclusion --- p.133 / Bibliography --- p.135

Thesis (Ph. D.) - University of Washington. / Bibliography: ℓ.[184]-185.

Thesis (Ph. D.)--Virginia Polytechnic Institute and State University, 1992. / Vita. Abstract. Includes bibliographical references (leaves 172-179). Also available via the Internet.

Consider a polyhedral surface in three-space that has the property that it can change its shape while keeping all its polygonal faces congruent. Adjacent faces are allowed to rotate along common edges. Mathematically exact flexible surfaces were found by Connelly in 1978. But the question remained as to whether the volume bounded by such surfaces was necessarily constant during the flex. In other words, is there a mathematically perfect bellows that actually will exhale and inhale as it flexes? For the known examples, the volume did remain constant. Following an idea of Sabitov, but using the theory of places in algebraic geometry (suggested by Steve Chase), Connelly et al. showed that there is no perfect mathematical bellows. All flexible surfaces must flex with constant volume. We built several models to illustrate the above theory, in particular, we built a model of the cubeoctahedron after a suggestion by Walser. This model is cut at a line of symmetry, pops up to minimize its energy stored by 4 rubber bands in its interior, and in doing so also maximizes its volume. Three MATLAB programs were written to illustrate how the cuboctahedron is obtained by truncation, how the physical cuboctahedron moves and how the motion of the cubeoctahedron can be described if self-intersection is possible.

No description available.

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.

In this thesis I study reconstruction of orthogonal polyhedral surfaces and orthogonal polyhedra from partial information about their boundaries. There are three main questions for which I provide novel results. The first question is "Given the dual graph, facial angles and edge lengths of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the dihedral angles?" The second question is "Given the dual graph, dihedral angles and edge lengths of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the facial angles?" The third question is "Given the vertex coordinates of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the edges and faces, possibly after rotating?" For the first two questions, I show that the answer is "yes" for genus-0 orthogonal polyhedra and polyhedral surfaces under some restrictions, and provide linear time algorithms. For the third question, I provide results and algorithms for orthogonally convex polyhedra. Many related problems are studied as well.

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.

In this thesis I study reconstruction of orthogonal polyhedral surfaces and orthogonal polyhedra from partial information about their boundaries. There are three main questions for which I provide novel results. The first question is "Given the dual graph, facial angles and edge lengths of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the dihedral angles?" The second question is "Given the dual graph, dihedral angles and edge lengths of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the facial angles?" The third question is "Given the vertex coordinates of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the edges and faces, possibly after rotating?" For the first two questions, I show that the answer is "yes" for genus-0 orthogonal polyhedra and polyhedral surfaces under some restrictions, and provide linear time algorithms. For the third question, I provide results and algorithms for orthogonally convex polyhedra. Many related problems are studied as well.

In the game of Kal-toh depicted in the television series Star Trek: Voyager, players attempt to create polyhedra by adding to a jumbled collection of metal rods. Inspired by this fictional game, we formulate graph-theoretical questions about polyhedral (triconnected and planar) subgraphs in an on-line environment. The problem of determining the existence of a polyhedral subgraph within a graph G is shown to be NP-hard, and we also give some non-trivial upper bounds for the problem of determining the minimum number of edge additions necessary to guarantee the existence of a polyhedral subgraph in G. A two-player formulation of Kal-toh is also explored, in which the first player to form a target subgraph is declared the winner. We show a polynomial-time solution for simple cases of this game but conjecture that the general problem is NP-hard.