A number theoretic question arising in the geometry of plane curves and in billiard dynamics

TL;DR

This paper proves a number theoretic property related to rational angles and tangent functions, with implications for the geometry of plane curves and billiard dynamics, showing certain equalities cannot occur for rational angles.

Contribution

It establishes a new number theoretic result linking rational angles and tangent functions, impacting the understanding of bicycle curves and billiard systems.

Findings

No integer n > 1 satisfies n tan(πρ) = tan(nπρ) for rational ρ ≠ 1/2.

Results have implications for the geometry of bicycle curves.

Findings contribute to the theory of mathematical billiards.

Abstract

We prove that if $\rho\neq1/2$ is a rational number between zero and one, then there is no integer $n>1$ such that $$ n\tan(\pi\rho)=\tan(n\pi\rho). $$ This has interpretations both in the theory of bicycle curves and that of mathematical billiards.