Given a Delaunay decomposition of a compact hyperbolic surface, one may record the topological data of the decomposition, together with the intersection angles between the `empty disks' circumscribing the regions of the decomposition. The main result of this paper is a characterization of when a given topological decomposition and angle assignment can be realized as the data of an actual Delaunay decomposition of a hyperbolic surface.
The Delaunay tessellation of a locally finite subset of hyperbolic space is constructed using convex hulls in Euclidean space of one higher dimension. For finite and lattice-invariant sets it is proven to be a polyhedral decomposition, and versions (necessarily modified from the Euclidean setting) of the empty circumspheres condition and geometric duality with the Voronoi tessellation are proved. Some pathological examples of infinite, non lattice-invariant sets are exhibited.
Motivated by recent work on Delaunay triangulations of hyperbolic surfaces, we consider the minimal number of vertices of such triangulations. First, we will show that every hyperbolic surface of genus $g$ has a simplicial Delaunay triangulation with $O(g)$ vertices, where edges are given by distance paths. Then, we will construct a class of hyperbolic surfaces for which the order of this bound is optimal. Finally, to give a general lower bound, we will show that the $Ω(\sqrt{g})$ lower bound for the number of vertices of a simplicial triangulation of a topological surface of genus $g$ is tigh…
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…
We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space. Thus the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles. More generally, w…
The Bolza surface can be seen as the quotient of the hyperbolic plane, represented by the Poincaré disk model, under the action of the group generated by the hyperbolic isometries identifying opposite sides of a regular octagon centered at the origin. We consider generalized Bolza surfaces $\mathbb{M}_g$, where the octagon is replaced by a regular $4g$-gon, leading to a genus $g$ surface. We propose an extension of Bowyer's algorithm to these surfaces. In particular, we compute the value of the systole of $\mathbb{M}_g$. We also propose algorithms computing small sets of points on $\mathbb{M}_…
We consider ``hyperideal'' circle patterns, i.e. patterns of disks appearing in the definition of the Delaunay decomposition associated to a set of disjoint disks, possibly with cone singularities at the center of those disks. Hyperideal circle patterns are associated to hyperideal hyperbolic polyhedra. We describe the possible intersection angles and singular curvatures of those circle patterns, on Euclidean or hyperbolic surfaces with conical singularities. This is related to results on the dihedral angles of ideal or hyperideal hyperbolic polyhedra.
We consider geometric triangulations of surfaces, i.e., triangulations whose edges can be realized by disjoint locally geodesic segments. We prove that the flip graph of geometric triangulations with fixed vertices of a flat torus or a closed hyperbolic surface is connected. We give upper bounds on the number of edge flips that are necessary to transform any geometric triangulation on such a surface into a Delaunay triangulation.
A non-trivial slope $r$ on a knot $K$ in $S^3$ is called a characterizing slope if whenever the result of $r$-surgery on a knot $K'$ is orientation preservingly homeomorphic to the result of $r$-surgery on $K$, then $K'$ is isotopic to $K$. Ni and Zhang ask: for any hyperbolic knot $K$, is a slope $r = p/q$ with $|p| + |q|$ sufficiently large a characterizing slope? In this article we answer this question in the negative by demonstrating that there is a hyperbolic knot $K$ in $S^3$ which has infinitely many non-characterizing slopes. As the simplest known example, the hyperbolic knot $8_6$ has…
We give sharp upper bounds on the maximal injectivity radius of finite-area hyperbolic surfaces and use them, for each g at least 2, to identify a constant r_{g-1,2} with the property that the set of closed genus-g hyperbolic surfaces with maximal injectivity radius at least r is compact if and only if r > r_{g-1,2}. The main tool is a version of the "centered dual complex" that we introduced earlier, a coarsening of the Delaunay complex of a locally finite set. In particular, we bound the area of a compact centered dual two-cell below given lower bounds on its side lengths.
Let X be a geodesic metric space. Gromov proved that there exists k>0 such that if every sufficiently large triangle T satisfies the Rips condition with constant k times pr(T), where pr(T) is the perimeter T, then X is hyperbolic. We give an elementary proof of this fact, also giving an estimate for k. We also show that if all the triangles T in X satisfy the Rips condition with constant k times pr(T), then X is a real tree. Moreover, we point out how this characterization of hyperbolicity can be used to improve a result by Bonk, and to provide an easy proof of the (well-known) fact that X is …
In this paper we characterize a function defined on the set of edges of a triangulated surface such that there is a spherical angle structure having the function as the edge invariant (or Delaunay invariant). We also characterize a function such that there is a hyperbolic angle structure having the function as the edge invariant.
Let $K\subset S^3$ be a hyperbolic fibered knot such that $S^3_{p/q}(K)$, the $\frac pq$--surgery on $K$, is non-hyperbolic. We prove that if the monodromy of $K$ is right-veering, then $0\le\frac pq\le 4g(K)$. The upper bound $4g(K)$ cannot be attained if $S^3_{p/q}(K)$ is a small Seifert fibered L-space. If the monodromy of $K$ is neither right-veering nor left-veering, then $|q|\le3$. As a corollary, for any given positive torus knot $T$, if $p/q\ge4g(T)+4$, then $p/q$ is a characterizing slope. This improves earlier bounds of Ni--Zhang and McCoy. We also prove that some finite/cyclic slope…
For every half-translation surface with marked points $(M,Σ)$, we construct an associated tessellation $Π(M,Σ)$ of the Poincaré upper half plane whose tiles have finitely many sides and area at most $π$. The tessellation $Π(M,Σ)$ is equivariant with respect to the action of $\mathrm{PSL}(2,\mathbb{R})$, and invariant with respect to (half-)translation covering. In the case $(M,Σ)$ is the torus $\mathbb{C}/\mathbb{Z}^2$ with a one marked point, $Π(\mathbb{C}/\mathbb{Z}^2,\{0\})$ coincides with the iso-Delaunay tessellation introduced by Veech as both tessellations give the Farey tessellation. A…
We prove that a 3--dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than $2π$ is the metric of the Gauss image of some convex polyhedral cusp. This result is an analog of the Rivin-Hodgson theorem characterizing compact convex hyperbolic polyhedra in terms of their Gauss images. The proof uses a variational method. Namely, a cusp with a given Gauss image is identified with a critical point of a…
Suppose $C$ is a compact, $n$-edged two-cell of the centered dual decomposition of a locally finite set in the hyperbolic plane, a coarsening of the Delaunay tessellation which was introduced in the author's prior work. We describe an effectively computable lower bound on the area of $C$, given an $n$-tuple of positive real numbers bounding the lengths of the edges of $C$ below. The ancillary materials contain Python code implementing (for $n<10$) an algorithm to compute this bound. For geometrically reasonable edge length bounds, we expect the given area bound to be sharp or near-sharp.
