Special Issue: Celebrating the 75th Birthday of V.V. Kozlov (Issue Editors: Sergey Bolotin, Vladimir Dragović, and Dmitry Treschev)

Poncelet Porism in Singular Cases

Vladimir Dragović^ab, Milena Radnović<sup>ca

a</sup> Mathematical Institute SANU, Belgrade, Kneza Mihaila 36, 11000 Belgrade, Serbia
^b The University of Texas at Dallas, Department of Mathematical Sciences, 800 W. Campbell Rd, 75080-3021 Richardson TX, USA
^c The University of Sydney, School of Mathematics and Statistics, Carslaw F07, 2006 NSW, Australia

Abstract: The celebrated Poncelet porism is usually studied for a pair of smooth conics that are in a general position. Here we discuss Poncelet porism in the real plane — affine or projective, when that is not the case, i. e., the conics have at least one point of tangency or at least one of the conics is not smooth. In all such cases, we find necessary and sufficient conditions for the existence of an $n$-gon inscribed in one of the conics and circumscribed about the other.

Keywords: Poncelet theorem, Cayley’s conditions, geometry of conics, elliptic curves, singular cubics, Chebyshev polynomials

MSC: 51N15, 14H70

Received: 30.04.2025
Accepted: 04.07.2025

Language: English

DOI: 10.1134/S1560354725040094