沿圆滑动的多边形的极值面积

Polygons with prescribed edge slopes: configuration space and extremal points of perimeter

We describe the configuration space $\mathbf{S}$ of polygons with prescribed edge slopes, and study the perimeter $\mathcal{P}$ as a Morse function on $\mathbf{S}$. We characterize critical points of $\mathcal{P}$ (these are \textit{tangential} polygons) and compute their Morse indices. This setup is motivated by a number of results about critical points and Morse indices of the oriented area function defined on the configuration space of polygons with prescribed edge lengths (flexible polygons). As a by-product, we present an independent computation of the Morse index of the area function (ob…

Polygon Vertex Extremality and Decomposition of Polygons

In this paper, we show that if we decompose a polygon into two smaller polygons, then by comparing the number of extremal vertices in the original polygon versus the sum of the two smaller polygons, we can gain at most two globally extremal vertices in the smaller polygons, as well as at most two locally extremal vertices. We then will derive two discrete Four-Vertex Theorems from our results.

沿曲线滑动的多边形的极值面积

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。

欧氏平面中多边形的一些极值问题

Some extremal problems for polygons in the Euclidean plane

本文研究了欧氏平面中凸多边形及其周长相关的极值问题，提出了若干表述简单但证明复杂的结果，并讨论了未解决的问题。

Extremal convex polygons inscribed in a given convex polygon

A convex polygon $Q$ is inscribed in a convex polygon $P$ if every side of $P$ contains at least one vertex of $Q$. We present algorithms for finding a minimum area and a minimum perimeter convex polygon inscribed in any given convex $n$-gon in $O(n)$ and $O(n^3)$ time, respectively. We also investigate other variants of this problem.

Extremal antipodal polygons and polytopes

Let $S$ be a set of $2n$ points on a circle such that for each point $p \in S$ also its antipodal (mirrored with respect to the circle center) point $p'$ belongs to $S$. A polygon $P$ of size $n$ is called \emph{antipodal} if it consists of precisely one point of each antipodal pair $(p,p')$ of $S$. We provide a complete characterization of antipodal polygons which maximize (minimize, respectively) the area among all antipodal polygons of $S$. Based on this characterization, a simple linear time algorithm is presented for computing extremal antipodal polygons. Moreover, for the generalization …