Areas of spherical and hyperbolic triangles in terms of their midpoints

TL;DR

This paper derives formulas relating the area of spherical and hyperbolic triangles to the midpoints of their sides, providing explicit relations involving determinants and scalar products.

Contribution

It introduces explicit formulas connecting triangle areas to side midpoints on spherical and hyperbolic surfaces, extending geometric relations in these non-Euclidean geometries.

Findings

Derived formulas for triangle area in terms of midpoints.

Established relations involving determinants and scalar products.

Clarified solution selection criteria for hyperbolic and spherical cases.

Abstract

Let $M$ be either the 2-sphere $\SS^2 \subset\RR^3$ or the hyperbolic plane $\HH^2 \subset \RR^3$. If $\Delta(abc)$ is a geodesic triangle on $M$ with corners at $a,b,c\in M$, we denote by $\alpha, \beta, \gamma\in M$ the midpoints of their sides. If $\Omega$ denotes the oriented area of this triangle on $M$, it satisfies the relations: $$ \sin(\Omega/2) = \frac{\Det_3(abc)}{\sqrt{2(1 + \scalar{a}b)(1 + \scalar{b}c)(1 + \scalar{c}a)}} = \Det_3(\alpha\beta\gamma) \, $$ where $\scalar{\}{\}$ denotes the Euclidean scalar product for $M=\SS^2$ and the Lorentzian scalar product for $M=\HH^2$. On the hyperbolic plane one should always take the solution with $\modu{\Omega/2}<\pi/2$. On the sphere, singular cases excepted, a straightforward procedure tells us which solution of this equation is the correct one.