[Paper Review] Centers of mass of Poncelet polygons, 200 years after

Richard Evan Schwartz, Serge Tabachnikov|arXiv (Cornell University)|Jul 16, 2016

Mathematics and Applications5 references3 citations

TL;DR

This paper investigates the loci of centers of mass of Poncelet polygons—closed polygons inscribed in one conic and circumscribed about another—200 years after Poncelet's original discovery. Using projective geometry and invariant measures, the authors prove that when the inner and outer conics are homothetic ellipses, the center of mass of the tangency points traces a single fixed point, generalizing Weill’s Theorem and revealing deep symmetry in Poncelet configurations.

ABSTRACT

The locus of the centers of mass of the family of Poncelet polygons, inscribed into a conic $Γ$ and circumscribed about a conic $γ$, is a conic homothetic to $Γ$.

Motivation & Objective

• To extend Poncelet’s closure theorem by analyzing the dynamics of centers of mass in Poncelet polygons.
• To investigate the geometric behavior of the center of mass of tangency points in Poncelet polygons inscribed in and circumscribed about nested ellipses.
• To prove that when the conics are homothetic, the center of mass of the tangency points remains fixed, generalizing Weill’s Theorem.
• To establish that the locus of the center of mass of the tangency polygon is an ellipse homothetic to the original conic under non-homothetic conditions.
• To explore the invariance of the center of mass under continuous deformation of Poncelet polygons via infinitesimal analysis.

Proposed method

• The authors use a cyclic coordinate system on the outer conic, parameterizing the Poncelet map as a translation in this coordinate, which relies on the invariance of the measure dt = dA / F(A), where F(A) is the tangent length from a point on the outer conic to the inner conic.
• They apply infinitesimal geometry to analyze displacements of tangency points, showing that the center of mass velocity vanishes under closed polygon constraints.
• The proof of Weill’s Theorem relies on the similarity of infinitesimal triangles formed by points on the outer circle and a fixed point, leading to the invariance of the tangent-length-normalized measure.
• The authors use the concept of polar duality to show that the tangency-point polygon Q_t is itself a Poncelet polygon with respect to a dual conic, enabling the analysis of its center of mass.
• They employ projective invariance techniques, though noting that centers of mass are not projectively invariant, requiring careful normalization.
• The analysis includes a historical reconstruction based on a 1814 letter from Konstantin Shestakov to Nikolai Lobachevsky, describing Poncelet’s original insights during captivity.

Experimental results

Research questions

• RQ1 Under what conditions does the center of mass of the tangency points of a Poncelet polygon remain fixed as the polygon rotates?
• RQ2 How does the locus of the center of mass of the tangency polygon Q_t behave when the outer and inner conics are not homothetic?
• RQ3 Can the invariance of the center of mass under continuous Poncelet motion be proven using infinitesimal geometry and invariant measures?
• RQ4 What is the geometric relationship between the center of mass of the tangency points and the original conics in the case of homothetic ellipses?
• RQ5 How does the historical account of Poncelet’s discovery in Saratov inform the modern mathematical treatment of Poncelet polygons?

Key findings

• When the inner and outer conics are homothetic ellipses, the center of mass of the tangency points of the Poncelet polygon remains fixed, proving Weill’s Theorem.
• For non-homothetic conics, the locus of the center of mass of the tangency points Q_t is an ellipse homothetic to the original conic γ.
• The center of mass of the vertices P_t of the Poncelet polygon traces a circle when the conics are concentric circles, but the locus becomes more complex under general configurations.
• The infinitesimal displacement of the center of mass vanishes due to the cancellation of vector sums over closed polygons, proving invariance under continuous motion.
• The measure dt = dA / F(A) is invariant under the Poncelet map, which allows the map to be expressed as a translation in the t-coordinate, ensuring periodicity independent of the starting point.
• The historical letter from Shestakov to Lobachevsky provides a primary source for Poncelet’s original geometric intuition, particularly the idea of a 'spinning polygon' that preserves its shape under rotation.