Areas of spherical and hyperbolic triangles in terms of their midpoints
arXiv:1307.2567
Abstract
Let be either the 2-sphere $\SS^2 \subset\RR^3$ or the hyperbolic plane $\HH^2 \subset \RR^3$. If is a geodesic triangle on with corners at , we denote by the midpoints of their sides. If denotes the oriented area of this triangle on , it satisfies the relations: $$ \sin(Ω/2) = \frac{\Det_3(abc)}{\sqrt{2(1 + \scalar{a}b)(1 + \scalar{b}c)(1 + \scalar{c}a)}} = \Det_3(αβγ) \, $$ 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 $\modu{Ω/2}<π/2$. On the sphere, singular cases excepted, a straightforward procedure tells us which solution of this equation is the correct one.
11 pages