The Inscribed Angle Theorem for the Hyperbola

TL;DR

This paper generalizes the inscribed angle theorem from circles to hyperbolas within Minkowski space, linking geometric properties to special relativity and providing insights into non-Euclidean geometries.

Contribution

It introduces a pseudo-angle concept in Minkowski space, extending classical circle theorems to hyperbolas and connecting these to relativistic and non-relativistic physics.

Findings

Generalization of inscribed angle theorem to hyperbolas in Minkowski space

Connection between pseudo-angles and special relativity

Limit case leading to an inscribed angle theorem for parabolas

Abstract

The inscribed angle theorem, a famous result about the angle subtended by a chord within a circle, is well known and commonly taught in school curricula. In this paper, we present a generalisation of this result (and other related circle theorems) to the rectangular hyperbola. The notion of angle is replaced by pseudo-angle, defined via the Minkowski inner product. Indeed, in Minkowski space, the unit hyperbola is the set of points a unit metric distance from the origin, analogous to the Euclidean unit circle. While this is a result of pure geometrical interest, the connection to Minkowski space allows an interpretation in terms of special relativity where, in the limit $c\to\infty$, it leads to a familiar result from non-relativistic dynamics. This non-relativistic result can be interpreted as an inscribed angle theorem for the parabola, which we show can also be obtained from the Euclidean inscribed angle theorem by taking the limit of a family of ellipses ananlogous to the non-relativistic limit $c\to\infty$. This simple result could be used as a pedagogical example to consolidate understanding of pseudo-angles in non-Euclidean spaces or to demonstrate the power of analytic continuation.