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Local overlaps in special unfoldings of convex polyhedra
 In Proc. 18th Canad. Conf. Comput. Geom
, 2006
"... We define a notion of local overlaps in polyhedron unfoldings. We use this concept to construct convex polyhedra for which certain classes of edge unfoldings contain overlaps, thereby negatively resolving some open conjectures. In particular, we construct a convex polyhedron for which every shortest ..."
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We define a notion of local overlaps in polyhedron unfoldings. We use this concept to construct convex polyhedra for which certain classes of edge unfoldings contain overlaps, thereby negatively resolving some open conjectures. In particular, we construct a convex polyhedron for which every shortest path unfolding contains an overlap. We also present a convex polyhedron for which every steepest edge unfolding contains an overlap. We conclude by analyzing a broad class of unfoldings and again find a convex polyhedron for which they all contain overlaps. 1
Cauchy’s Theorem and Edge Lengths of Convex Polyhedra
"... In this paper we explore, from an algorithmic point of view, the extent to which the facial angles and combinatorial structure of a convex polyhedron determine the polyhedron—in particular the edge lengths and dihedral angles of the polyhedron. Cauchy’s rigidity theorem of 1813 states that the dihe ..."
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In this paper we explore, from an algorithmic point of view, the extent to which the facial angles and combinatorial structure of a convex polyhedron determine the polyhedron—in particular the edge lengths and dihedral angles of the polyhedron. Cauchy’s rigidity theorem of 1813 states that the dihedral angles are uniquely determined. Finding them is a significant algorithmic problem which we express as a spherical graph drawing problem. Our main result is that the edge lengths, although not uniquely determined, can be found via linear programming. We make use of significant mathematics on convex polyhedra by Stoker, Van Heijenoort, Gale, and Shepherd.
EdgeUnfolding AlmostFlat Convex Polyhedral Terrains
, 2013
"... In this thesis we consider the centuriesold question of edgeunfolding convex polyhedra, focusing specifically on edgeunfoldability of convex polyhedral terrain which are “almost flat ” in that they have very small height. We demonstrate how to determine whether cuttrees of such almostflat terra ..."
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In this thesis we consider the centuriesold question of edgeunfolding convex polyhedra, focusing specifically on edgeunfoldability of convex polyhedral terrain which are “almost flat ” in that they have very small height. We demonstrate how to determine whether cuttrees of such almostflat terrains unfold and prove that, in this context, any partial cuttree which unfolds without overlap and “opens ” at a root edge can be locally extended by a neighboring edge of this root edge. We show that, for certain (but not all) planar graphs G, there are cuttrees which unfold for all almostflat terrains whose planar projection is G. We also demonstrate a noncuttreebased method of unfolding which relies on “slice ” operations to build an unfolding of a complicated terrain from a known unfolding of a simpler terrain. Finally, we describe several heuristics for generating cutforests and provide some computational results of such heuristics on unfolding almostflat convex polyhedral terrains.