The geometry of cyclic hyperbolic polygons

Abstract

A hyperbolic polygon is defined to be cyclic, horocyclic, or equidistant if its vertices lie on a metric circle, horocycle, or a component of the equidistant locus to a hyperbolic geodesic, respectively. Convex such n-gons are parametrized by the subspaces of (R + ) n that contain their side length collections, and area and circumcircle or "collar" radius determine symmetric, smooth functions on these spaces. We give formulas for and bounds on the derivatives of these functions, and make some observations on their behavior. Notably, the monotonicity properties of area and circumcircle radius exhibit qualitative differences on the collection of centered vs non-centered cyclic polygons, where a cyclic polygon is centered if it contains the center of its circumcircle in its interior.An n-tuple (d 0 , . . . , d n−1 ) of positive real numbers is the side length collection of a compact convex n-gon P in the hyperbolic plane H 2 that is cyclic or horocyclic, or has all vertices equidistant from a fixed geodesic, as long as d i ≤ j =i d j for each i. Moreover, such a polygon P is uniquely prescribed up to orientation-preserving isometry of H 2 by this n-tuple up to cyclic permutation. It is cyclic if and only if for each i,and P is horocyclic if and only if for some i equality holds above.This fact seems to have been independently rediscovered several times. It was recorded by the late W.W. Stothers [13], then by J.M. Schlenker (see [12, p. 2175]) as part of a larger project. Schlenker's results on cyclic polygons were re-proved by R. Walter [15], [16]. Propositions 1.8, 3.4 and 4.6 here combine for a self-contained case-by-case proof. Our approach is standard; compare eg. Robbins [11] (in the Euclidean setting). All aspects are elementary, but the cyclic case is somewhat subtle; see the discussion at the beginning of Section 1.Our main goal here is to make some observations on the qualitative behavior of area and circumcircle radius, thought of as functions on a subset AC n of (R + ) n that parametrizes cyclic n-gons by side length, for each n ≥ 3. We also record relatively simple formulas for their derivatives, and give useful bounds in some cases. In view of the facts above we define:One of our key themes is that the functions on AC n determined by area and circumcircle radius exhibit qualitatively different behaviors on centered versus non-centered cyclic n-gons.This paper was mostly written during a period of partial NSF support. 1 arXiv:1101.4971v5 [math.GT] 30 Jun 2015 Hyperbolic trigonometry relates the distance between points on a circle to the angle from one to another and the circle radius. Here is the key function: Lemma 1.4. For d > 0 and J ≥ d/2, let T be a hyperbolic triangle with two sides of length J and a third of length d. Its interior angle at the vertex opposite the side of length d is: θ(d, J) = 2 sin −1 (sinh(d/2)/ sinh J) ∈ [0, π] This is a continuous function on {(d, J) | 0 < d ≤ 2J}, smooth on its interior. For fixed d > 0, θ(d, J) decreases in J on [d/2, ∞), with θ(d, d/2) = π, lim J→∞ θ(d, J) ...

The local maxima of maximal injectivity radius among hyperbolic surfaces

“…If the coefficient ∂ ∂dn D 0 (T i + n (d(t))) + D 0 (T i − n (d(t))) is non-zero then we can solve for d n (t), yielding a first-order ODE in d n (t). We claim this holds at t = 0, ie for the T i ± n (d), and therefore at all possible values of d(t) near d. Given the claim, Picard's theorem on the existence of solutions to first-order ODE implies there is a smooth solution d n (t) for small t (note that smoothness of D 0 is proven in [4]). We will apply results from Section 2 of [3], together with [4, Proposition 2.3], to prove the claim.…”

“…Second, by construction Delaunay cells are cyclic or horocyclic; that is, inscribed in metric circles or horocycles, respectively. In [4] there are calculus formulas describing the derivative of area with respect to side length for such polygons.…”

“…Here for each j, T i + j and T i − j are the triangles containing the edge γ j ; for a triangle T i with edges γ j 1 , γ j 2 , γ j 3 we refer by T i (d(t)) to the triple (d j 1 (t), d j 2 (t), d j 3 (t)) of changing edge lengths; and D 0 (a, b, c) records the area of the triangle with edge lengths a, b and c. In [4] we gave formulas for the partial derivatives of D 0 with respect to a, b and c.…”

The centered dual and the maximal injectivity radius of hyperbolic surfaces

“…. , b k−1 ) ∈ [0, ∞) k−1 , since f (b 1 ,...,b k−1 ) (x) → 0 as x → ∞ there is an unbounded interval consisting of possible choices of b k for which equation (4) has no solutions. In the remainder of this section, we will analyze the case of three-holed spheres using an elementary direct method, providing a corrected version of the theorem in this case.…”

Bounds for several-disk packings of hyperbolic surfaces

“…Analogously, two horocyclic ideal triangles with the same compact side length are isometric (this is easy to prove bare hands, or cf. [10,Prop. 3.4]), and every one has a reflection exchanging its two vertices in H 2 .…”