# On curves with the Poritsky property

@article{Glutsyuk2019OnCW, title={On curves with the Poritsky property}, author={A. A. Glutsyuk}, journal={Journal of Fixed Point Theory and Applications}, year={2019}, volume={24}, pages={1-60} }

Reflection in planar billiard acts on oriented lines. For a given closed convex planar curve $$\gamma $$ γ , the string construction yields a one-parameter family $$\Gamma _p$$ Γ p of nested billiard tables containing $$\gamma $$ γ for which $$\gamma $$ γ is a caustic: the reflection from $$\Gamma _p$$ Γ p sends each tangent line to $$\gamma $$ γ to a line tangent to $$\gamma $$ γ . The reflections from $$\Gamma _p$$ Γ p act on the corresponding tangency points, inducing a family of string…

## 12 Citations

### On curves with the Poritsky property

- Materials ScienceJournal of Fixed Point Theory and Applications
- 2022

Reflection in planar billiard acts on oriented lines. For a given closed convex planar curve γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}…

### On infinitely many foliations by caustics in strictly convex open billiards

- Mathematics
- 2021

Reﬂection in strictly convex bounded planar billiard acts on the space of oriented lines and preserves a standard area form. A caustic is a curve C whose tangent lines are reﬂected by the billiard to…

### On infinitely many foliations by caustics in strictly convex non-closed billiards

- Mathematics
- 2021

Reflection in strictly convex bounded planar billiard acts on the space of oriented lines and preserves a standard area form. A caustic is a curve C whose tangent lines are reflected by the billiard…

### Four equivalent properties of integrable billiards

- MathematicsIsrael Journal of Mathematics
- 2021

By a classical result of Darboux, a foliation of a Riemannian surface has the Graves property (also known as the strong evolution property) if and only if the foliation comes from a Liouville net. A…

### The Ballet of Triangle Centers on the Elliptic Billiard

- Physics
- 2020

A bevy of new phenomena relating to (i) the shape of 3-periodics and (ii) the kinematics of certain Triangle Centers constrained to the Billiard boundary are explored, specifically the non-monotonic motion some can display with respect to 3- periodics.

### The geometry of billiards in ellipses and their poncelet grids

- MathematicsJournal of Geometry
- 2021

The goal of this paper is an analysis of the geometry of billiards in ellipses, based on properties of confocal central conics. The extended sides of the billiards meet at points which are located on…

### Remarks on Joachimsthal Integral and Poritsky Property

- Mathematics
- 2020

The billiard in an ellipse has a conserved quantity, the Joachimsthal integral. We show that the existence of such an integral characterizes conics. We extend this result to the spherical and…

### On the motion of billiards in ellipses

- MathematicsEuropean Journal of Mathematics
- 2022

For billiards in an ellipse e with an ellipse as caustic, there exist canonical coordinates on e such that the billiard transformation from vertex to vertex is equivalent to a shift of coordinates. A…

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By a classical result of Darboux, a foliation of a Riemannian surface has the Graves property (also known as the strong evolution property) if and only if the foliation comes from a Liouville net. A…

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The billiard in an ellipse has a conserved quantity, the Joachimsthal integral. We show that the existence of such an integral characterizes conics. We extend this result to the spherical and…