Euclidean Structures on Simplicial Surfaces and Hyperbolic Volume

Rivin, Igor

doi:10.2307/2118572

Cited by 212 publications

(276 citation statements)

References 5 publications

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“…The first proof that an outerplanar graph can be realized as a Delaunay triangulation relies on two elegant results due to Dillencourt and Smith [3] and to Rivin [4,5] Next, we need this result of Rivin [4,5]. To complete our first proof, we just need to show that:…”

Section: The First Proofmentioning

confidence: 99%

“…Consider a face in G D corresponding to such vertex in G. For each expansion of the backbone edge, the weight of the cycle edge of that face is decremented by /2. Similarly, for each expansion, the weight of each of the two cycle edges is increased by 4 . It remains to prove that the weight of each edge of G D satisfies condition 2a of Theorem 5, if lies within a certain range.…”

Section: Expansionmentioning

confidence: 99%

“…We prove this first for the extreme case where a face f of G D shares all the backbone edge, as in Fig 4. First we show the upper bound of w(e). Each time a backbone edge is expanded, the weight of each of the two cycle edges which are the edges of two adjacent faces of f is increased by 4 . Therefore, each such edge takes extra at most 4 (k + 1) from the weights, where k is the number of backbone edges in G D or chords in G .…”

Section: Lemmamentioning

confidence: 99%

“…The first one is an easy consequence of Dillencourt and Smith's [3] criterion relating Hamiltonianicity and inscribability. The second one, which occupies most of this note, uses Rivin's [4] inscribability criterion and constructs an explicit "witness" of this inscribability, in the form of certain weights assigned to the edges of the graph.…”

mentioning

confidence: 99%

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Outerplanar Graphs and Delaunay Triangulations

Alam

Rivin

Streinu

2012

Computation, Physics and Beyond

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Section: The First Proofmentioning

confidence: 99%

Section: Expansionmentioning

confidence: 99%

Section: Lemmamentioning

confidence: 99%

mentioning

confidence: 99%

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Outerplanar Graphs and Delaunay Triangulations

Alam

Rivin

Streinu

2012

Computation, Physics and Beyond

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“…Rivin [18,19] presents a polynomial time test for Delaunay realizability that is based on ideas in hyperbolic geometry. In contrast, the HMS test is based on elementary Euclidean geometric ideas.…”

Section: The Hms Test For Delaunay Realizabilitymentioning

confidence: 99%

On the Efficiency of a Local Iterative Algorithm to Compute Delaunay Realizations

Lillis

Pemmaraju

Experimental Algorithms

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Abstract. Greedy routing protocols for wireless sensor networks (WSNs) are fast and efficient but in general cannot guarantee message delivery. Hence researchers are interested in the problem of embedding WSNs in low dimensional space (e.g., R 2 ) in a way that guarantees message delivery with greedy routing. It is well known that Delaunay triangulations are such embeddings. We present the algorithm FindAngles, which is a fast, simple, local distributed algorithm that computes a Delaunay triangulation from any given combinatorial graph that is Delaunay realizable. Our algorithm is based on a characterization of Delaunay realizability due to Hiroshima et al. (IEICE 2000). When compared to the PowerDiagram algorithm of Chen et al. (SoCG 2007) that embeds triangulations in the plane so as to permit successful greedy routing, our algorithm requires on average 1/6 th the number of iterations. FindAngles also scales linearly to larger networks and has a much faster distributed implementation than PowerDiagram, requiring just a single round of communication in each iteration. The PowerDiagram algorithm was proposed as an improvement on another algorithm due to Thurston (unpublished, 1988). Our experiments show that on average the PowerDiagram algorithm uses about 19% fewer iterations than the Thurston algorithm, whereas our algorithm uses about 89% fewer iterations. Experimentally, FindAngles exhibits well behaved convergence. Theoretically, we prove that with certain initial conditions the error term decreases monotonically. Taken together, these suggest our algorithm may have polynomial time convergence for certain classes of graphs. We note that our algorithm runs only on Delaunay realizable triangulations. This is not a significant concern because Hiroshima et al. (IEICE 2000) indicate that most combinatorial triangulations are indeed Delaunay realizable, which we have also observed experimentally: out of 5000 randomly generated combinatorial triangulations on 100 vertices, only one was not Delaunay realizable.

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Quadrangulations of sphere and ball quotients

Zeytin

Uludağ

2013

Math. Nachr.

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We give a classification of sphere quadrangulations satisfying a condition of non‐negative curvature, following Thurston's classification of sphere triangulations under the same condition. The generic family of quadrangulations is parametrized by the points of positive square‐norm of an integral Gaußian lattice Λ′ in the six‐dimensional complex Lorentz space. There is a subgroup of automorphisms of Λ′ acting on this lattice whose orbits parametrize sphere quadrangulations in a one‐to‐one manner. This group acts discretely on the corresponding five‐dimensional complex hyperbolic space; is of finite co‐volume; its ball quotient is the moduli space of unordered 8 points on the Riemann sphere, and also appears in Picard‐Terada‐Deligne‐Mostow list. Both Thurston's lattice and our lattice may be thought of as parametrizations of certain families of subgroups of the modular group; equivalently, of certain families of dessins. These families also parametrize points of a moduli space.

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Euclidean Structures on Simplicial Surfaces and Hyperbolic Volume

Cited by 212 publications

References 5 publications

Outerplanar Graphs and Delaunay Triangulations

Outerplanar Graphs and Delaunay Triangulations

On the Efficiency of a Local Iterative Algorithm to Compute Delaunay Realizations

Quadrangulations of sphere and ball quotients

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