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Poncelet theorems

W. Barth, Th. Bauer

TL;DR

This work presents a unified, modern treatment of Poncelet-type theorems by recasting classical closing problems in projective geometry as translations on elliptic curves or as rational degenerations. Starting from a three-dimensional space Poncelet theorem for pairs of quadrics, the authors derive plane and conic analogues, count inscribed conics in pencils, and extend to configurations of three conics and circles, all via elliptic-curve dynamics and involutive correspondences. They provide explicit, computable Poncelet-n conditions for revolution-quadric pairs, develop circle geometry through circle invariants, and connect to classical results (Emch, Steiner, Money-Coutts) through a common elliptic-curve framework. The approach highlights deep links between projective geometry, elliptic curves, and integrable systems, offering compact proofs and new perspective on closing phenomena in diverse geometric settings.

Abstract

If there is one polygon inscribed into some smooth conic and circumscribed about another one, then there are infinitely many such polygons. This is Poncelet's theorem. The aim of this note is to collect some (mostly classical) versions of this theorem, namely: - Weyr's Poncelet theorem in $P_3$ (1870), - Emch's theorem on circular series (1901), - Gerbaldi's formula for the number of Poncelet pairs (1919), - the Money-Coutts theorem on three circles (1971), - the zig-zag theorem (1974), - a (probably new) Poncelet theorem on three conics, - a Poncelet formula for quadrics of revolution.

Poncelet theorems

TL;DR

This work presents a unified, modern treatment of Poncelet-type theorems by recasting classical closing problems in projective geometry as translations on elliptic curves or as rational degenerations. Starting from a three-dimensional space Poncelet theorem for pairs of quadrics, the authors derive plane and conic analogues, count inscribed conics in pencils, and extend to configurations of three conics and circles, all via elliptic-curve dynamics and involutive correspondences. They provide explicit, computable Poncelet-n conditions for revolution-quadric pairs, develop circle geometry through circle invariants, and connect to classical results (Emch, Steiner, Money-Coutts) through a common elliptic-curve framework. The approach highlights deep links between projective geometry, elliptic curves, and integrable systems, offering compact proofs and new perspective on closing phenomena in diverse geometric settings.

Abstract

If there is one polygon inscribed into some smooth conic and circumscribed about another one, then there are infinitely many such polygons. This is Poncelet's theorem. The aim of this note is to collect some (mostly classical) versions of this theorem, namely: - Weyr's Poncelet theorem in (1870), - Emch's theorem on circular series (1901), - Gerbaldi's formula for the number of Poncelet pairs (1919), - the Money-Coutts theorem on three circles (1971), - the zig-zag theorem (1974), - a (probably new) Poncelet theorem on three conics, - a Poncelet formula for quadrics of revolution.

Paper Structure

This paper contains 10 sections, 23 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Weyr's Poncelet theorem in ${\rm I\!P}_3$
  3. Poncelet pairs in a pencil
  4. Gerbaldi's formula for the number of inscribed conics
  5. A Poncelet theorem on three conics
  6. The Poncelet condition for quadrics of revolution
  7. Circles in the projective plane
  8. Emch's and Steiner's theorems on circular series
  9. The zig-zag theorem
  10. The Money-Coutts theorem

Key Result

Theorem 1.1

Suppose that there exists a closed sequence of distinct lines $L_1,\dots,L_{2n},L_{2n+1}=L_1$ such that the line $L_i$ belongs to $R_1$ resp. $R_2$, if $i$ is odd resp. even, and such that consecutive lines $L_i,L_{i+1}$ intersect each other. Then there are such closed sequences of length $2n$ throu

Theorems & Definitions (23)

  • Theorem 1.1: Weyr p. 28, Hurw p. 13
  • Proposition 1.3
  • Proposition 2.1
  • Corollary 2.2
  • Theorem 3.1: Ger19, p. 103
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 4.1
  • Proposition 5.2
  • Proposition 6.1: Ped57, p. 32
  • ...and 13 more