2025/01/18 by Dragović, Vladimir, Murad, Mohammad Hassan
#14H70 #34M55 #37A10 #37J70 #51N20 #Algebraic Geometry (math.AG) #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2501.10795
We study pairs of conics (D,P), called n-Poncelet pairs, such that an n-gon, called an n-Poncelet polygon, can be inscribed into D and circumscribed about P. Here D is a circle and P is a parabola from a confocal pencil F with the focus F. We prove that the circle contains F if and only if every parabola P\inF forms a 3-Poncelet pair with the circle. We prove that the center of D coincides with F if and only if every parabola P∈ F forms a 4-Poncelet pair with the circle. We refer to such property, observed for n=3 and n=4, as n-isoperiodicity. We prove that F is not n-isoperiodic with any circle D for n different from 3 and 4. Using isoperiodicity, we construct explicit algebraic solutions to Painlevé VI equations.