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Computing Signed Permutations of Polygons
, 2002
"... Given a planar polygon (or chain) with a list of edges fe 1 ; e 2 ; e 3 ; : : : ; e n 1 ; e n g, we examine the effect of several operations that permute this edge list, resulting in the formation of a new polygon. The main operations that we consider are: reversals which involve inverting the order ..."
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Cited by 6 (4 self)
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Given a planar polygon (or chain) with a list of edges fe 1 ; e 2 ; e 3 ; : : : ; e n 1 ; e n g, we examine the effect of several operations that permute this edge list, resulting in the formation of a new polygon. The main operations that we consider are: reversals which involve inverting the order of a sublist, transpositions which involve interchanging subchains (sublists), and edgeswaps which are a special case and involve interchanging two consecutive edges. Using these permuting operations, we explore the complexity of performing certain actions, such as convexifying a given polygon or obtaining its mirror image. When each edge of the given polygon has also been assigned a parity we say that the polygon is signed. In this case any edge involved in a reversal changes parity. The complexity of some problems varies depending on whether a polygon is signed or unsigned. An additional restriction in many cases is that polygons remain simple after every permutation.
EdgeUnfolding Nested Polyhedral Bands
, 2006
"... A band is the intersection of the surface of a convex polyhedron with the space between two parallel planes, as long as this space does not contain any vertices of the polyhedron. The intersection of the planes and the polyhedron produces two convex polygons. If one of these polygons contains the ot ..."
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Cited by 4 (3 self)
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A band is the intersection of the surface of a convex polyhedron with the space between two parallel planes, as long as this space does not contain any vertices of the polyhedron. The intersection of the planes and the polyhedron produces two convex polygons. If one of these polygons contains the other in the projection orthogonal to the parallel planes, then the band is nested. We prove that all nested bands can be unfolded, by cutting along exactly one edge and folding continuously to place all faces of the band into a plane, without intersection.
Unfolding Prismatoids as Convex Patches: Counterexamples and Positive Results
, 2012
"... We address the unsolved problem of unfolding prismatoids in a new context, viewing a “topless prismatoid” as a convex patch—a polyhedral subset of the surface of a convex polyhedron homeomorphic to a disk. We show that several natural strategies for unfolding a prismatoid can fail, but obtain a pos ..."
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Cited by 2 (1 self)
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We address the unsolved problem of unfolding prismatoids in a new context, viewing a “topless prismatoid” as a convex patch—a polyhedral subset of the surface of a convex polyhedron homeomorphic to a disk. We show that several natural strategies for unfolding a prismatoid can fail, but obtain a positive result for “petal unfolding ” topless prismatoids. We also show that the natural extension to a convex patch consisting of a face of a polyhedron and all its incident faces, does not always have a nonoverlapping petal unfolding. However, we obtain a positive result by excluding the problematical patches. This then leads a positive result for restricted prismatoids. Finally, we suggest suggest studying the unfolding of convex patches in general, and offer some possible lines of investigation.
Zipper Unfolding of Domes and Prismoids
, 2013
"... We study Hamiltonian unfolding—cutting a convex polyhedron along a Hamiltonian path of edges to unfold it without overlap—of two classes of polyhedra. Such unfoldings could be implemented by a single zipper, so they are also known as zipper edge unfoldings. First we consider domes, which are simple ..."
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Cited by 1 (0 self)
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We study Hamiltonian unfolding—cutting a convex polyhedron along a Hamiltonian path of edges to unfold it without overlap—of two classes of polyhedra. Such unfoldings could be implemented by a single zipper, so they are also known as zipper edge unfoldings. First we consider domes, which are simple convex polyhedra. We find a family of domes whose graphs are Hamiltonian, yet any Hamiltonian unfolding causes overlap, making the domes Hamiltonianununfoldable. Second we turn to prismoids, which are another family of simple convex polyhedra. We show that any nested prismoid is Hamiltonianunfoldable, and that for general prismoids, Hamiltonian unfoldability can be tested in polynomial time.
Band Unfoldings and Prismatoids: A Counterexample
, 2007
"... This note shows that the hope expressed in [ADL + 07]—that the new algorithm for edgeunfolding any polyhedral band without overlap might lead to an algorithm for unfolding any prismatoid without overlap—cannot be realized. A prismatoid is constructed whose sides constitute a nested polyhedral band, ..."
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This note shows that the hope expressed in [ADL + 07]—that the new algorithm for edgeunfolding any polyhedral band without overlap might lead to an algorithm for unfolding any prismatoid without overlap—cannot be realized. A prismatoid is constructed whose sides constitute a nested polyhedral band, with the property that every placement of the prismatoid top face overlaps with the band unfolding.