Notes on Beltrami's Essay

TL;DR

This paper revisits Beltrami's 1868 work on hyperbolic geometry, providing detailed derivations of key results like hyperbolic distance, triangle angle sums, and equations for geometric figures.

Contribution

It offers a comprehensive derivation of Beltrami's principal results, clarifying his equations and proofs in hyperbolic geometry.

Findings

Derivation of hyperbolic distance formula on the disc

Proof that hyperbolic triangle angles sum to less than 180°

Equations for circles, equidistants, and horocycles in hyperbolic geometry

Abstract

Eugenio Beltrami published his seminal 'Essay on the Interpretation of Non-Euclidean Geometry' in 1868, where he showed that geodesics on a surface of constant negative curvature can be mapped as straight lines on a Euclidean disc. More importantly he showed that figures on the disc would satisfy the identities of hyperbolic geometry characteristic of a surface of negative curvature. However Beltrami did not always give a full explanation of the equations which he used. These notes are an attempt to provide a derivation of some of his principal results, including his formula for hyperbolic distance on the disc, his proof that the sum of the (hyperbolic) angles of a triangle on the disc is less than two right angles and his equations for circles, equidistants and horocycles.