Benjamin Aaron Bailey
Consider the Poincare disc model for hyperbolic geometry. In this paper, a convenient computational formula is developed along with an aesthetic geometric interpretation. Two proofs, one geometric and one analytical, of each result are given.
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This second part on polygons in the hyperbolic plane is based on the first part which deals with uniqueness and existence of cocyclic polygons with prescribed sidelengths. The topic here is the maximum question for the area of these polygons. It is shown that the maximal copies must be cocyclic and oriented-convex or have a special collinear shape. The maximal copies are unique up to rigid hyperbolic motions. The cocyclicity means that the vertices are situated on a distance circle, a distance line or a horocycle, the three types of cycles in the hyperbolic plane. The phenomenon of different c…
On the area of ordinary hyperbolic reduced polygons
本文研究双曲平面中约化凸体的面积问题，针对一类双曲约化n边形，证明正n边形在该族中面积最大。
There are many four vertex type theorems appearing in the literature, coming in both smooth and discrete flavors. The most familiar of these is the classical theorem in differential geometry, which states that the curvature function of a simple smooth closed curve in the plane has at least four extreme values. This theorem admits a natural discretization to Euclidean polygons due to O. Musin. In this article we adapt the techniques of Musin and prove a discrete four vertex theorem for convex hyperbolic polygons.
On the perimeter, diameter and circumradius of ordinary hyperbolic reduced polygons
本文研究双曲平面中的普通约化多边形，回答了 Lassak 关于其周长、直径和外接圆半径的问题，并将 Fabińska 的相关结果推广到双曲情形。
The hexagon is the least-perimeter tile in the Euclidean plane. On hyperbolic surfaces, the isoperimetric problem differs for every given area. Cox conjectured that a regular $k$-gonal tile with 120-degree angles is isoperimetric for its area. We prove his conjecture and more.
After work of W. P. Thurston, C. Bavard and É. Ghys constructed particular hyperbolic polyhedra from spaces of deformations of Euclidean polygons. We present this construction as a straightforward consequence of the theory of mixed-volumes. The gluing of these polyhedra can be isometrically embedded into complex hyperbolic cone-manifolds constructed by Thurston from spaces of deformations of Euclidean polyhedra. It is then possible to deduce the metric structure of the spaces of polygons embedded in complex hyperbolic orbifolds discovered by P. Deligne and G. D. Mostow.
We prove that every rational angled hyperbolic triangle has transcendental side lengths and that every rational angled hyperbolic quadrilateral has at least one transcendental side length. Thus, there does not exist a rational angled hyperbolic triangle or quadrilateral with algebraic side lengths. We conjecture that there does not exist a rational angled hyperbolic polygon with algebraic side lengths.
Certain topics on polygons are extended from Euclidean to hyperbolic geometry. This first part deals with uniqueness and existence of cocyclic polygons with prescribed sidelengths. The non-Euclidean versions are more difficult due to the existence of three different types of circles in the hyperbolic plane. The second part will be concerned with the problem of maximizing the area of polygons with fixed sidelengths. By the metric character of these problems it is necessary to dispense to a certain extent with classical hyperbolic trigonometry because angles must be replaced by length quantities…
Reduced polygons in the hyperbolic plane
本文研究双曲平面中约化多边形，通过厚度概念刻画其极小性，并利用厚度估计其直径，揭示该类几何对象的性质。
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 $(0,\infty)^n$ 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 …
Formulas about the side lengths, diagonal lengths or radius of the circumcircle of a cyclic polygon in Euclidean geometry, hyperbolic geometry or spherical geometry can be unified.
Extremal Area of Polygons sliding along Curves
本文研究多边形顶点沿曲线滑动时的面积函数，给出临界点几何判据及Hesse矩阵，为Morse理论方法奠基，并引出内面积台球这一新概念。
T. Keleti asked, whether the ratio of the perimeter and the area of a finite union of unit squares is always at most 4. In this paper we present an example where the ratio is greater than 4. We also answer the analogous question for regular triangles negatively and list a number of open problems.
A Geometric proof for the Polygonal Isoperimetric Inequality
本文利用约束问题的最优性条件与三角形几何基本思想，从几何角度证明了在给定面积下最小化周长的多边形必定是正多边形。
Symmedians as Hyperbolic Barycenters
该研究提出类似中线的新型动态坐标化方法，将其推广为理想双曲多边形的双曲重心并证明其保留相似性质，同时开启相关模空间及额外最优性的研究。
Heron's formula states that the area $K$ of a triangle with sides $a$, $b$, and $c$ is given by $$ K=\sqrt {s(s-a) (s-b) (s-c)} $$ where $s$ is the semiperimeter $(a+b+c)/2$. Brahmagupta, Robbins, Roskies, and Maley generalized this formula for polygons of up to eight sides inscribed in a circle. In this paper we derive formulas giving the areas of any $n$-gon, with odd $n$, in terms of the ordered list of side lengths, if the $n$-gon is circumscribed about a circle (instead of being inscribed in a circle). Unlike the cyclic polygon problem, where the order of the sides does not matter, for th…
Hedgehog Reconstruction of Polygons: Non-Central Sections and Slabs
本文提出通过非中心截面和板面积重建多边形的方法，证明了其唯一性及中心对称多边形的重建可能性。
It is known that cyclic configurations of a planar polygonal linkage are critical points of the signed area function. In the paper, we announce an explicit formula of the Morse index for the signed area of a cyclic configuration. It depends not only on the combinatorics of a cyclic configuration, but also includes some metric characterization.
We consider a hyperbolic polygon in the unit disk $\{z:\, |z|<1\}$ with all its vertices on the unit circle $\{z:\, |z|=1\}$ and a growth process of such polygons when each $n$-gon generates an $n(n-1)$-gon by inverting itself across all of its sides. In this paper, we prove some general monotonicity results of inversion for convex hyperbolic $n$-gons and solve an extremal problem that, among all convex hyperbolic $4$-gons containing the origin, the inverted side length of the longest side of the given hyperbolic $4$-gon is minimal for the regular hyperbolic $4$-gon.
This chapter is motivated by the paper by Thurston on triangulations of the sphere and singular flat metrics on the sphere. Thurston locally parametrized the moduli space of singular flat metrics on the sphere with prescribed positive curvature data by the complex hyperbolic space of appropriate dimension. This work can be considered as a generalization of signature calculation of the Hermitian form that he made in his paper. The moduli space of singular flat metrics having unit area on the sphere with prescribed curvature data can be locally parametrized by certain spaces of polygons. This ca…
All-angles bisection of polygon area
研究如何找到给定方向上平分简单多边形面积的直线方程及以最少计算改变方向，针对‘平分凸’多边形给出解决方案。
We confirm two conjectures of Lassak on the area of reduced spherical polygons. The area of every reduced spherical non-regular $n$-gon is less than that of the regular spherical $n$-gon of the same thickness. Moreover, the area of every reduced spherical polygon is less than that of the regular spherical odd-gons of the same thickness and whose number of vertices tends to infinity.
We investigate the ``generalized Heron polynomial'' that relates the squared area of an n-gon inscribed in a circle to the squares of its side lengths. For a (2m+1)-gon or (2m+2)-gon, we express it as the defining polynomial of a certain variety derived from the variety of binary (2m-1)-forms having m-1 double roots. Thus we obtain explicit formulas for the areas of cyclic heptagons and octagons, and illuminate some mysterious features of Robbins' formulas for the areas of cyclic pentagons and hexagons. We also introduce a companion family of polynomials that relate the squared area of an n-go…
An algorithm to find maximum area polygons circumscribed about a convex polygon
本文提出一种O(n^3)时间算法，用于寻找外切于任意给定凸n边形的最大面积凸多边形，并以此推翻了Farris猜想，同时给出了正n边形情形的显式解。
This paper presents a general method for obtaining radius of the corresponding circumference to a cyclical polygon $n$ sides given the lengths of said sides, using the notion of complex number. As of radius $r$, obtained, can then be calculated polygon area in question applying known expressions which requires, after $n=4$, a powerful calculation tool. Are also given like elements regarding non convex polygons cyclic, although in this respect it deepens less and is not considered, course, the calculation of areas.
Tight bounds on the maximal area of small polygons: Improved Mossinghoff polygons
针对偶数顶点小多边形最大面积未知的问题，构造了面积为单变量函数最大值的新小n边形，证明其面积优于Mossinghoff构造的最优结果，且n=6时达到最大面积。
Two ideal polygons, $(p_1,\ldots,p_n)$ and $(q_1,\ldots,q_n)$, in the hyperbolic plane or in hyperbolic space are said to be $α$-related if the cross-ratio $[p_i,p_{i+1},q_i,q_{i+1}] = α$ for all $i$ (the vertices lie on the projective line, real or complex, respectively). For example, if $α= -1$, the respective sides of the two polygons are orthogonal. This relation extends to twisted ideal polygons, that is, polygons with monodromy, and it descends to the moduli space of Möbius-equivalent polygons. We prove that this relation, which is, generically, a 2-2 map, is completely integrable in the…
Amicable Lattice Rhombuses are Amicable
本文研究了相亲菱形的性质，证明了所有相亲的整数格菱形实际上都是等量的。
In his works [R1,R2] David Robbins proposed several interrelated conjectures on the area of the polygons inscribed in a circle as an algebraic function of its sides. Most recently, these conjectures have been established in the course of several independent investigations. In this note we give an informal outline of these developments.
Holmes-Thompson area of inscribed polygons and convex projective structures
本文研究了正向旗在三维空间中定义的凸多边形，并探讨了霍姆斯-汤普森面积与希尔伯特度量之间的关系，以及有限面积凸实投影结构的参数化问题。
Isoperimetric inequality for non-Euclidean polygons
针对正多边形在固定周长与边数下面积最大这一经典等周结论，本文将其推广到非欧情形，并完整证明了双曲几何与球面几何中的多边形等周不等式。
A polygon is \textit{small} if it has unit diameter. The maximal area of a small polygon with a fixed number of sides $n$ is not known when $n$ is even and $n\geq14$. We determine an improved lower bound for the maximal area of a small $n$-gon for this case. The improvement affects the $1/n^3$ term of an asymptotic expansion; prior advances affected less significant terms. This bound cannot be improved by more than $O(1/n^3)$. For $n=6$, $8$, $10$, and $12$, the polygon we construct has maximal area.
Given the n vertices of a convex polygon in cyclic order, can the triangle of maximum area inscribed in P be determined by an algorithm with O(n) time complexity? A purported linear-time algorithm by Dobkin and Snyder from 1979 has recently been shown to be incorrect by Keikha, Löffler, Urhausen, and van der Hoog. These authors give an alternative algorithm with O(n log n) time complexity. Here we give an algorithm with linear time complexity.
We study the locus of the Circumcenter of Mass of Poncelet polygons, and the limit of the Center of Mass (when we consider the polygon as a "homogeneous lamina") for degenerate Poncelet polygons. We also provide a proof for one of Dan Reznik invariants for billiard trajectories. Plus, we take a look at how the scene looks like when we shift to spherical geometry.
Extremal Area of Polygons, sliding along a Circle
本文研究了顶点在圆或椭圆上的多边形面积函数的临界构型，计算了孤立临界点的莫尔斯指数，并将退化点与特征值问题关联，揭示了偶数维中的非孤立奇点结构。
We say that a polygon inscribed in the circle is asymmetric if it contains no two antipodal points being the endpoints of a diameter. Given $n$ diameters of a circle and a positive integer $k<n$, this paper addresses the problem of computing a maximum area asymmetric $k$-gon having as vertices $k<n$ endpoints of the given diameters. The study of this type of polygons is motivated by ethnomusiciological applications.
Polygons of unit area with vertices in sets of infinite planar measure
研究探讨了面积为1的多边形顶点在无限测度集合中的存在性，分别回答了循环四边形和等边凸多边形的问题。
Sequences of polygons generated by performing iterative processes on an initial polygon have been studied extensively. One of the most popular sequences is the one sometimes referred to as {\it Kasner polygons}. Given a polygon $K$, the first Kasner descendant $K'$ of $K$ is obtained by placing the vertices of $K'$ at the midpoints of the edges of $K$. More generally, for any fixed $m$ in $(0,1)$ one may define a sequence of polygons $\{K^{t}\}_{t\ge 0}$ where each polygon $K^{t}$ is obtained by dividing every edge of $K^{t-1}$ into the ratio $m:(1-m)$ in the counterclockwise (or clockwise) di…
We describe a hyperbolic version of the Ambartzumian-Pleijel identity. We use this identity to prove the hyperbolic Crofton formula and the hyperbolic isoperimetric inequality. This identity also provides a way to compute the chord length distribution for an ideal polygon in the hyperbolic plane. The analogous results for a maximally symmetric, simply connected, $2$-dimensional Riemannian manifold with constant sectional curvature are given at the end.
A central limit theorem for random disc-polygons in smooth convex discs
本文证明了在光滑凸盘中随机圆多边形面积的定量中心极限定理，采用Stein方法和Fodor等人证明的方差渐近下界。
On random disc-polygons in a disc-polygon
研究凸多边形内随机圆多边形的顶点数和缺失面积的渐进行为，提出r-凸的类比结果。
Area bounds for planar convex bodies containing a fixed number of interior integral points
本文研究包含固定数量内部整点的平面凸体的面积界问题，利用格宽数据证明面积界，并给出有理多边形依赖于分母的精确面积界及基于诺特公式的格多边形面积方程。
Outer length billiards on polygons
本文研究多边形台面上的外长度台球系统，描述远离台面的轨道行为，并对正方形台面的逃逸轨道提出猜想，补充了处理光滑正曲率台面的相关研究。
We study hyperbolic polyhedral surfaces with faces isometric to regular hyperbolic polygons satisfying that the total angles at vertices are at least $2π.$ The combinatorial information of these surfaces is shown to be identified with that of Euclidean polyhedral surfaces with negative combinatorial curvature everywhere. We prove that there is a gap between areas of non-smooth hyperbolic polyhedral surfaces and the area of smooth hyperbolic surfaces. The numerical result for the gap is obtained for hyperbolic polyhedral surfaces, homeomorphic to the double torus, whose 1-skeletons are cubic gr…
Given a convex $n$-gon $P$ and a positive integer $m$ such that $3\le m\le n-1$, let $Q$ denote the largest area convex $m$-gon contained in $P$. We are interested in the minimum value of $Δ(Q)/Δ(P)$, the ratio of the areas of these two polygons. More precisely, given positive integers $n$ and $m$, with $3 \le m \le n-1$, define \begin{equation*} f_n(m)=\min_{P\in \mathcal {P}_n} \max_{Q \subset P,|Q|=m} \frac{Δ(Q)}{Δ(P)} \end{equation*} where the maximum is taken over all $m$-gons contained in $P$, and the minimum is taken over $\mathcal{P}_n$, the entire class of convex $n$-gons. The values …
In Groeneboom (1988) a central limit theorem for the number of vertices of the convex hull of a uniform sample from the interior of convex polygon was derived. In the unpublished preprint Nagaev and Khamdamov (1991) (in Russian) a central limit result for the joint distribution of the number of vertices and the remaining area is given, using a coupling of the sample process near the border of the polygon with a Poisson point process as in Groeneboom (1988), and representing the remaining area in the Poisson approximation as a union of a doubly infinite sequence of independent standard exponent…
On triangulations with fixed areas
本文证明固定面积的三角剖分数量有限，且代数顶点下等分剖分具有代数性质。
New variational arguments regarding the Blaschke-Lebesgue theorem
本文通过计算Reuleaux多边形面积对顶点扰动的敏感性，证明正则多边形是唯一的临界多边形，从而为Blaschke-Lebesgue和Firey-Sallee定理提供了新的变分证明。
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