Igor Pak
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.
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An almost forgotten gem of Gauss tells us how to compute the area of a pentagon by just going around it and measuring areas of each vertex triangles (i.e. triangles whose vertices are three consecutive vertices of the pentagon). We give several proofs and extensions of this beautiful formula to hexagon etc. and consider special cases of affine--regular polygons. The Gauss pentagon formula is, in fact, equivalent to the Monge formula which is equivalent to the Ptolemy formula. On the other hand, we give a new proof of the Robbins formula for the area of a cyclic pentagon in terms of the side le…
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.
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.
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.
We characterize the topological configurations of points and lines that may arise when placing n points on a circle and drawing the n perpendicular bisectors of the sides of the corresponding convex cyclic n-gon. We also provide exact and asymptotic formulas describing a random realizable configuration, obtained either by sampling the points uniformly at random on the circle or by sampling a realizable configuration uniformly at random.
It is well known that Heron's theorem provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its sides. It has been extended by Brahmagupta to quadrilaterals inscribed in a circle (cyclic quadrilaterals). A natural problem is trying to further generalize the result to cyclic polygons with a larger number of edges, which, surprisingly, has revealed to be far from simple. In this paper we investigate such a problem by following a new and elementary approach. We start from the simple observation that the incircle of a right triangle touches its hypothen…
We provide a new proof of the elementary geometric theorem on the existence and uniqueness of cyclic polygons with prescribed side lengths. The proof is based on a variational principle involving the central angles of the polygon as variables. The uniqueness follows from the concavity of the target function. The existence proof relies on a fundamental inequality of information theory. We also provide proofs for the corresponding theorems of spherical and hyperbolic geometry (and, as a byproduct, in $1+1$ spacetime). The spherical theorem is reduced to the euclidean one. The proof of the hyperb…
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 …
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…
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…
If we fix the angles at the vertices of a convex planar $n$-gon, the lengths of its edges must satisfy two linear constraints in order for it to close up. If we also require unit perimeter, our vectors of $n$ edge lengths form a convex polytope of dimension $n-3$, each facet of which consists of those $n$-gons in which the length of a particular edge has fallen to zero. Bavard and Ghys require unit area instead, which gives them a hyperbolic polytope. Those two polytopes are combinatorially equivalent, so either is fine for our purposes. Such a fixed-angles polytope is combinatorially richer w…
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。
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.
The Kernel Deficit Dominates Twice the Hull Deficit: A Sharp Strengthening of Nakano's Inequality
该研究强化了中野不等式，证明多边形核亏空主导两倍凸包亏空，通过凸体帽并集等方法推导，结果有Lean 4机器验证。
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