Rational Hyperbolic Triangles and a Quartic Model of Elliptic Curves

TL;DR

This paper explores the connection between rational hyperbolic triangles and a quartic model of elliptic curves, extending classical Euclidean triangle relationships to hyperbolic geometry and constructing examples with specific properties.

Contribution

It develops a hyperbolic analog of the elliptic curve model for Euclidean triangles using a quartic curve, enabling the construction of rational hyperbolic triangles with prescribed inradius and perimeter.

Findings

Hyperbolic triangles correspond to points on a quartic curve with genus 1.

The quartic curve can be realized as an intersection of two quadrics, allowing point addition.

Constructed examples of rational hyperbolic triangles with given inradius and perimeter.

Abstract

The family of Euclidean triangles having some fixed perimeter and area can be identified with a subset of points on a nonsingular cubic plane curve, i.e., an elliptic curve; furthermore, if the perimeter and the square of the area are rational, then the curve has rational coordinates and those triangles with rational side lengths correspond to rational points on the curve. We first recall this connection, and then we develop hyperbolic analogs. There are interesting relationships between the arithmetic on the elliptic curve (rank and torsion) and the family of triangles living on it. In the hyperbolic setting, the analogous plane curve is a quartic with two singularities at infinity, so the genus is still 1. We can add points geometrically by realizing the quartic as the intersection of two quadric surfaces. This allows us to construct nontrivial examples of rational hyperbolic triangles having the same inradius and perimeter as a given rational right hyperbolic triangle.