Hyperbolae are the locus of constant angle difference

TL;DR

This paper reveals that points on a hyperbola maintain a constant difference in angles at two fixed points, extending the classical focus-distance property of hyperbolae to an angle-based geometric property.

Contribution

The paper introduces a novel geometric property of hyperbolae, showing they preserve a constant angle difference at two fixed points, which was not previously well-known.

Findings

Hyperbolae maintain a constant angle difference at two fixed points.

This property extends the classical focus-distance characteristic of hyperbolae.

The result is motivated by recent work on Voronoi diagrams of turning rays.

Abstract

Given two points A,B in the plane, the locus of all points P for which the angles at A and B in the triangle A,B,P have a constant sum is a circular arc, by Thales' theorem. We show that the difference of these angles is kept a constant by points P on a hyperbola (albeit with foci different from A and B). Whereas hyperbolae are well-known to maintain a constant difference between the distances to their foci, the above angle property seems not to be widely known. The question was motivated by recent work by Alegr\'ia et al. and De Berg et al. on Voronoi diagrams of turning rays.