A dynamical system in the space of convex quadrangles

TL;DR

This paper investigates a dynamical system defined on the space of convex quadrangles with fixed angles and perimeter, exploring how a specific transformation based on edge lengths and angles behaves.

Contribution

It introduces a novel dynamical system on convex quadrangles with fixed angles and perimeter, analyzing its properties and behavior.

Findings

Characterization of the fixed points of the system

Analysis of stability and convergence properties

Insights into the geometric transformations involved

Abstract

Let us consider a family $F(\alpha,\beta,\gamma,\delta)$ of convex quadrangles in the plane with given angles $\{\alpha,\beta,\gamma,\delta\}$ and with the perimeter $2\pi$. Such quadrangle $Q\in F(\alpha,\beta,\gamma,\delta)$ can be considered as a point $(x_1,x_2,x_3,x_4)\in\mathbb{R}^4$, where $\{x_1,x_2,x_3,x_4\}$ are lengths of edges. Then to $F$ a finite open segment $I\subset\mathbb{R}^4$ is corresponded. A quadrangle in $F$, that corresponds to the midpoint of $I$ is called a \emph{balanced quadrangle}. Let $M$ be the set of balanced quadrangles. The function $f:M\to M$ is defined in the following way: angles of the balanced quadrangle $Q'$, $Q'=f(Q)$, are numerically equal to edges of $Q$. The map $f$ defines a dynamical system in the space of balanced quadrangles. In this work we study properties of this system.