Poncelet's theorem and dual billiards
type: Item
instance of: scholarly article
title: Poncelet's theorem and dual billiards
zbMATH Open document ID: 0804.58020
published in: L'Enseignement Mathématique. 2e Série
publication date: 1994-06-29T00:00:00Z
review text: A short proof, based on a construction of a special area form and a dual billiard transformation, of the classical Poncelet's theorem is established. The statement is as follows. Given two nested ellipses \(\Gamma_ 0\) and \(\Gamma_ 1\), \(\Gamma_ 0 \subset \text{interior}(\Gamma_ 1)\) we choose a point \(x \in \Gamma_ 1\), draw a tangent line to \(\Gamma_ 0\) from \(x\), find its intersection \(y\) with \(\Gamma_ 1\) and iterate, taking \(y\) as a new starting point. The claim is that, if \(x\) returns back to the initial position after a finite number of iterations, then every point of \(\Gamma_ 1\) will return back after the same number of iterations.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 37C85
zbMATH DE Number: 597687
zbMATH Keywords: Lagrangian foliation
zbMATH Keywords: dual billiard
zbMATH Keywords: Poncelet's theorem
author: S. Tabachnikov
reviewed by: Jerzy Ombach
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 597687