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16
Greedy optimal homotopy and homology generators
 Proc. 16th Ann. ACMSIAM Symp. Discrete Algorithms
, 2005
"... Abstract We describe simple greedy algorithms to construct the shortest set of loops that generates either the fundamental group (with a given basepoint) or the first homology group (over any fixed coefficient field) of any oriented 2manifold. In particular, we show that the shortest set of loops t ..."
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Cited by 97 (11 self)
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Abstract We describe simple greedy algorithms to construct the shortest set of loops that generates either the fundamental group (with a given basepoint) or the first homology group (over any fixed coefficient field) of any oriented 2manifold. In particular, we show that the shortest set of loops that generate the fundamental group of any oriented combinatorial 2manifold, with any given basepoint, can be constructed in O(n log n) time using a straightforward application of Dijkstra's shortest path algorithm. This solves an open problem of Colin de Verdi`ere and Lazarus.
Unfolding convex polyhedra via quasigeodesics: Abstract
 In Proc. 17th Annu. Fall Workshop Comput. Comb. Geom., November 2007. [IOV09] [IV08a] [IV08b] Jinichi Itoh, Joseph O’Rourke, and Costin
"... We show that cutting shortest paths from every vertex of a convex polyhedron to a simple closed quasigeodesic, and cutting all but a short segment of the quasigeodesic, unfolds the surface to a planar simple polygon. 1 ..."
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Cited by 5 (4 self)
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We show that cutting shortest paths from every vertex of a convex polyhedron to a simple closed quasigeodesic, and cutting all but a short segment of the quasigeodesic, unfolds the surface to a planar simple polygon. 1
Inflating polyhedral surfaces
, 2006
"... Abstract. We prove that all polyhedral surfaces in R 3 have volumeincreasing isometric deformations. This resolves the conjecture of Bleecker who proved it for convex simplicial surfaces [B1]. A version of this result is proved for all convex surfaces in R d. We also discuss limits on the volume of ..."
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Cited by 3 (0 self)
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Abstract. We prove that all polyhedral surfaces in R 3 have volumeincreasing isometric deformations. This resolves the conjecture of Bleecker who proved it for convex simplicial surfaces [B1]. A version of this result is proved for all convex surfaces in R d. We also discuss limits on the volume of such deformations, present a number of conjectures and special cases.
Star Unfolding Convex Polyhedra via Quasigeodesic Loops
, 2009
"... We extend the notion of star unfolding to be based on a quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple (nonoverlapping), planar polygon: cut along one shortest path from each vertex of P to Q, and cut all but ..."
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Cited by 2 (0 self)
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We extend the notion of star unfolding to be based on a quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple (nonoverlapping), planar polygon: cut along one shortest path from each vertex of P to Q, and cut all but one segment of Q.
HOW TO CUT OUT A CONVEX POLYHEDRON
, 2009
"... It is known that one can fold a convex polyhedron from a nonoverlapping face unfolding, but the complexity of the algorithm in [MP] remains an open problem. In this paper we show that every convex polyhedron P ⊂ R d can be obtained in polynomial time, by starting with a cube which contains P and ..."
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Cited by 2 (0 self)
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It is known that one can fold a convex polyhedron from a nonoverlapping face unfolding, but the complexity of the algorithm in [MP] remains an open problem. In this paper we show that every convex polyhedron P ⊂ R d can be obtained in polynomial time, by starting with a cube which contains P and sequentially cutting out the extra parts of the surface. Our main tool is of independent interest. We prove that given a convex polytope P in R d and a facet F of P, then F is contained in the union ∪G=F AG. Here the union is over all the facets G of P different from F, and AG is the set obtained from G by rotating towards F the hyperplane spanned by G about the intersection of it with the hyperplane spanned by F.
Continuous Blooming of Convex Polyhedra
, 2009
"... We construct the first two continuous bloomings of all convex polyhedra. First, the source unfolding can be continuously bloomed. Second, any unfolding of a convex polyhedron can be refined (further cut, by a linear number of cuts) to have a continuous blooming. ..."
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Cited by 1 (0 self)
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We construct the first two continuous bloomings of all convex polyhedra. First, the source unfolding can be continuously bloomed. Second, any unfolding of a convex polyhedron can be refined (further cut, by a linear number of cuts) to have a continuous blooming.
Continuous Unfolding of Polyhedra – a Motion Planning Approach
"... Abstract — Cut along the surface of a polyhedron and unfold it to a planar structure without overlapping is known as Unfolding Polyhedra problem which has been extensively studied in the mathematics literature for centuries. However, whether there exists a continuous unfolding motion such that the p ..."
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Abstract — Cut along the surface of a polyhedron and unfold it to a planar structure without overlapping is known as Unfolding Polyhedra problem which has been extensively studied in the mathematics literature for centuries. However, whether there exists a continuous unfolding motion such that the polyhedron can be continuously transformed to its unfolding has not been well studied. Recently, researchers started to recognize continuous unfolding as a key step in designing and implementation of selffolding robots. In this paper, we model the unfolding of a polyhedron as multilink treestructure articulated robot, and address this problem using motion planning techniques. Instead of sampling in continuous domain which traditional motion planners do, we propose to sample only in the discrete domain. Our experimental results show that sampling in discrete domain is efficient and effective for finding feasible unfolding paths. I.