Several combinatorial aspects of triangulations and their generalizations are studied in this thesis. A triangulation of a point configuration and a d-dimensional polyhedron whose vertices are among the points is a decomposition of the polyhedron using d-simplices with vertices among the points. The two main fields triangulations appear are combinatorial geometry in mathematics and computational geometry in information science. The topics connected to triangulations in combinatorial geometry include, polytope theory, Grobner bases of affine toric ideals, Hilbert bases, generalized hypergeometric functions, and Ehrhart polynomials. Many fields of computational geometry, such as computer graphics, solid modeling, mesh generation, and motion planning, use triangulations extensively. The problems we consider are among the main interests in combinatorial geometry. The topics are on simplified basic situations for computational geometry, but are those arising in applications and giving advice for planning algorithms. The objects we study are those around triangulations: (1) triangulations and dissections of 3-polytopes, (2) triangulations (mainly) in the plane, (3) 2-dimensional spheres, and (4) 2-dimensional simplicial complexes. They are from rigid to abstract in this order. The properties of these objects we are interested
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611
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Algorithms in combinatorial geometry
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73
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Deformable free space tilings for kinetic collision detection
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68
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Constructions and complexity of secondary polytopes
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65
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Combinatorial Convexity and Algebraic Geometry
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64
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An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere
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How to draw a planar graph on a grid. Combinatorica
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An algorithmic theory of lattice points in polyhedra
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Shellable decompositions of cells and spheres
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48
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On the definition and construction of pockets in macromolecules
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47
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The polytope of all triangulations of a point configuration
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39
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Sufficiency of McMullen’s conditions for f-vectors of simplicial polytopes
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37
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Unsolved Problems in Geometry
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36
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Dynamic ray shooting and shortest paths in planar subdivision via balanced geodesic triangulations
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34
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Discriminants, resultants and multidimensional determinants
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34
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On the Complexity of Some Basic Problems in Computational Convexity: II. Volume and Mixed Volumes
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Topology and Geometry
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28
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An acyclicity theorem for cell complexes in d dimensions
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22
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Tetrahedrizing point sets in three dimensions
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18
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A proof of the strict monotone 4−step conjecture, Contemporary Mathematics 223
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17
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Which spheres are shellable
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15
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An enumeration of simplicial 4-polytopes with 8 vertices
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14
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Hilbert bases, unimodular triangulations, and binary covers of rational polyhedral cones
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14
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A simple and relatively efficient triangulation of the n-cube
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13
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12
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A representation of 2-dimensional pseudomanifolds and its use in the design of a linear-time shelling algorithm
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11
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Simplexity of the cube
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8
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Provan, A decomposition property for simplicial complexes and its relation to diameters and shellings
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8
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6
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Triangulations (tilings) and certain block triangular matrices
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5
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Combinatorics of constructible complexes
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4
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de Fraysseix and Patrice Ossona de Mendez, On topological aspects of orientations
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2
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shooting in polygons using geodesic triangulations. Algorithmica 12
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2
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org Rambau, Francisco Santos, The Cayley trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings
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unter M.Ziegler, Deformed products and maximal shadows of polytopes, Contemporary Mathematics 223
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urgen Richter-Gebert, Minimal simplicial dissections and triangulations of convex 3-polytopes, Discrete and
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urgen Richter-Gebert, Finding minimal triangulations of convex 3-polytopes is NP-hard
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Spellman Munson, Triangulations of oriented matroids and convex polytopes
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Kimmo Eriksson, Extendable shellability for rank 3 matroid complexes, Discrete Math
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Gubeladze, Ng o Vi et Trung Normal polytopes, triangulations and Koszul algebras
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Fumihiko Takeuchi, Extremal properties for dissections of convex 3-polytopes
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Ernst Peter M ucke, Three-dimensional alpha shapes
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