The geometry of cyclic hyperbolic polygons

DeBlois, Jason

doi:10.48550/arxiv.1101.4971

Cited by 2 publications

(54 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Assume now that C v is not centered and apply [7,Lemma 1.5]. This produces an edge γ of C v and a half-space H containing C v and bounded by the geodesic containing γ, such that v is in the half-space H opposite H. [7, Lemma 1.5] further asserts that P ∪ T (e, v) is a convex polygon; also, γ is the unique longest edge of C v , by [7,Corollary 1.11]. We claim that the other endpoint w of the geometric dual e to γ is further from H than v, and hence that e is non-centered with initial vertex v.…”

Section: The Centered Dual To the Voronoi Tessellationmentioning

confidence: 99%

“…. , γ i , for each i the triangle T i described in the hypothesis of [7,Lemma 1.6] is identical to ∆(e i , v), where e i is the geometric dual to γ i . If C v is non-centered then Lemma 2.5 implies that γ v as defined above is its unique longest edge, so [7,Corollary 1.11] and [7,Lemma 1.5]…”

Section: 2mentioning

confidence: 99%

“…There is no corresponding result for cyclic polygons (at least no good one) because of a nonmonotonicity property of the area of those that are non-centered. See Section 3.1 below, where we will collect useful results from [7] on cyclic polygons.…”

Section: Admissible Spaces and Area Bounds: The Compact Casementioning

confidence: 99%

“…Given this it is natural to parametrize the set of cyclic n-gons by n-tuples of real numbers that satisfy this inequality. Here we will record some differential formulas that we proved in [7], treating the area and radius of cyclic polygons as functions on this space.…”

Section: Admissible Spaces and Area Bounds: The Compact Casementioning

confidence: 99%

“…This runs parallel to Section 3 of [5], but the definitions are modified to accommodate non-compact Voronoi edges. The fact that motivates our definition is that among cyclic polygons in H 2 , increasing the length of an edge increases area if and only if that edge is not the longest of a non-centered polygon, see [7]. Here is the definition of a "centered" polygon.…”

mentioning

confidence: 99%

See 4 more Smart Citations

The centered dual and the maximal injectivity radius of hyperbolic surfaces

DeBlois

2015

Geom. Topol.

Self Cite

View full text Add to dashboard Cite

We give sharp upper bounds on the injectivity radii of complete hyperbolic surfaces of finite area with some geodesic boundary components. The given bounds are over all such surfaces with any fixed topology; in particular, boundary lengths are not fixed. This extends the first author's earlier result to the with-boundary setting. In the second part of the paper we comment on another direction for extending this result, via the systole of loops function.THE MAXIMAL INJECTIVITY RADIUS OF HYPERBOLIC SURFACES WITH GEODESIC BOUNDARY 23

show abstract

Section: The Centered Dual To the Voronoi Tessellationmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Admissible Spaces and Area Bounds: The Compact Casementioning

confidence: 99%

Section: Admissible Spaces and Area Bounds: The Compact Casementioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

The centered dual and the maximal injectivity radius of hyperbolic surfaces

DeBlois

2015

Geom. Topol.

Self Cite

View full text Add to dashboard Cite

show abstract

Tessellations of hyperbolic surfaces

DeBlois

2011

Preprint

Self Cite

View full text Add to dashboard Cite

A finite subset S of a closed hyperbolic surface F canonically determines a centered dual decomposition of F : a cell structure with vertex set S, geodesic edges, and 2-cells that are unions of the corresponding Delaunay polygons. Unlike a Delaunay polygon, a centered dual 2-cell Q is not determined by its collection of edge lengths; but together with its combinatorics, these determine an admissible space parametrizing geometric possibilities for the Delaunay cells comprising Q. We illustrate its application by using the centered dual decomposition to extract combinatorial information about the Delaunay tessellation among certain genus-2 surfaces, and with this relate injectivity radius to covering radius here.A finite subset S of a closed hyperbolic surface F canonically determines a Voronoi tessellation V and Delaunay tessellation P , polygonal decompositions of F that are dual in a certain sense. Let us briefly outline this construction. Fix a locally isometric universal covering π : H 2 → F and let S = π −1 (S). The Voronoi tessellation V of H 2 determined by S is a cell complex structure where each x ∈ S determines a polygonal 2-cell V x defined by:x ∈ S, y ∈ S − {x}}, and each point of V (0) is equidistant from at least 3 points of S (see Section 1). The geometric dual to an edge V x ∩ V y of V is the geodesic arc γ xy in H 2 joining x to y, and the set of geometric duals to edges of V is the edge set of the Delaunay tessellation P determined by S. The covering action of π 1 F on H 2 leaves V and P invariant, and these descend to the tessellations V and P of F . P and V are dual in the sense that their edge sets are canonically bijective, as is the vertex set of each with the face set of the other. However, in some cases an edge e = V x ∩ V y of V is not centered (see Definition 3.1): int e does not intersect the geometric dual γ xy ⊂ P to e. We will regard this as a pathology of P , and "fix" it with the centered dual decomposition. Before we outline this construction, here is a sample application of our methods: Theorem 0.1. Let r β = d β /2 > 0, where cosh d β is the real root of x 3 − 14x 2 − 15x − 4. The Delaunay tessellation of a closed, orientable hyperbolic surface F of genus 2 determined by {x} has all edges centered if F has injectivity radius r ≥ r β at x. It is a triangulation unless r = r β and each edge has length d β , in which case it has a single quadrilateral 2-cell.The numerical value of r β is roughly 1.7006, whereas Boröczky's theorem [1] implies a universal upper bound of r α ∼ = 1.7191 on the injectivity radius of a genus-2 hyperbolic surface at a point x (see Lemma 2.3). Example 2.2 describes a surface F α with injectivity radius r α Partially supported by NSF grant DMS-1007175.

show abstract

The geometry of cyclic hyperbolic polygons

Cited by 2 publications

References 0 publications

The centered dual and the maximal injectivity radius of hyperbolic surfaces

The centered dual and the maximal injectivity radius of hyperbolic surfaces

Tessellations of hyperbolic surfaces

Contact Info

Product

Resources

About