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21
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 106 (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.
A geometric approach to the global attractor conjecture
 SIAM J. Appl. Dyn. Syst
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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.
Duality structures and discrete conformal variations of piecewise constant curvature surfaces
"... Abstract. A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating that for a chosen background geometry (Euclidean, hyperbolic, or spherical), each simplex has an isometric embedding into the background geometry w ..."
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Cited by 3 (1 self)
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Abstract. A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating that for a chosen background geometry (Euclidean, hyperbolic, or spherical), each simplex has an isometric embedding into the background geometry with the chosen edge lengths. Additional structure is defined either by giving a geometric structure to the Poincare ́ dual of the triangulation or by assigning a discrete metric, a way of assigning length to oriented edges. This notion leads to a notion of discrete conformal structure, generalizing the discrete conformal structures based on circle packings and their generalizations studied by Thurston and others. We define and analyze conformal variations of piecewise constant curvature 2manifolds, giving particular attention to the variation of angles. We give formulas for the derivatives of angles in each background geometry, which yield formulas for the derivatives of curvatures. Our formulas allow us to identify particular curvature functionals associated with conformal variations. Finally, we provide a complete classification of discrete conformal structures in each of the background geometries. 1.
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 3 (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.