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Circular Isoptics in Flatland

Alexander Thomas

TL;DR

The paper studies convex sets $S$ in the plane for which the angle of sight from every point on a fixed circle $C$ to $S$ is constant, introducing the notion of isoptics $S_\alpha$. It provides a dynamical-systems formulation on $C$ showing that $T^{2}$ acts as a rotation by $2\alpha$, which yields a clean dichotomy between irrational and rational multiples of $\pi$. It proves a sharp classification: nontrivial constant-angle shapes exist if and only if $\alpha=\frac{p}{q}\pi$ with $(p,q)=1$ and $q-p$ odd; for irrational $\alpha$ only disks occur, while in the rational case non-disk examples arise via perturbations of a regular $2q$-gon, accompanied by a billiards interpretation. The work connects to Green's classical results on constant-width-type shapes, introduces outer constant-angle billiards, and outlines a rich set of open questions and contest-related directions for further research.

Abstract

We explore convex shapes $S$ in the Euclidean plane which have the following property: there is a circle $C$ such that the angle between the two tangents from any point of $C$ to $S$ is constant equal to $α$. A dynamical formulation allows to analyze the existence of such shapes. Interestingly, the existence of non-circular shapes depends in a non-trivial way on the angle $α$.

Circular Isoptics in Flatland

TL;DR

The paper studies convex sets in the plane for which the angle of sight from every point on a fixed circle to is constant, introducing the notion of isoptics . It provides a dynamical-systems formulation on showing that acts as a rotation by , which yields a clean dichotomy between irrational and rational multiples of . It proves a sharp classification: nontrivial constant-angle shapes exist if and only if with and odd; for irrational only disks occur, while in the rational case non-disk examples arise via perturbations of a regular -gon, accompanied by a billiards interpretation. The work connects to Green's classical results on constant-width-type shapes, introduces outer constant-angle billiards, and outlines a rich set of open questions and contest-related directions for further research.

Abstract

We explore convex shapes in the Euclidean plane which have the following property: there is a circle such that the angle between the two tangents from any point of to is constant equal to . A dynamical formulation allows to analyze the existence of such shapes. Interestingly, the existence of non-circular shapes depends in a non-trivial way on the angle .

Paper Structure

This paper contains 4 sections, 5 theorems, 8 figures.

Table of Contents

  1. Flatland
  2. Dynamical viewpoint
  3. A strange billiards
  4. Opening

Key Result

Theorem 1.3

There exist shapes of constant angle $\alpha$ other than the disk if and only if $\alpha=\tfrac{p}{q}\pi$, where $(p,q)$ are two coprime intergers with $p<q$ and $q-p$ is odd. In that case, there are uncountably many examples.

Figures (8)

  • Figure 1.1: Angles of sight
  • Figure 1.2: Reuleaux triangle
  • Figure 1.3: Orthoptic circle
  • Figure 2.1: Dynamical viewpoint
  • Figure 2.2: Examples of periodic trajectories, $\alpha=\tfrac{\pi}{3}$ (left) and $\alpha=\tfrac{\pi}{6}$ (right).
  • ...and 3 more figures

Theorems & Definitions (12)

  • Theorem 1.3: Green 1949
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Example 2.7
  • ...and 2 more