Define M k 2 as the 2-dimensional metric space of constant curvature k . So, for example, M 0 2 is the Euclidean plane, M 1 2 is the surface of the unit sphere, and M − 1 2 is the hyperbolic plane.
Let X be a metric space. Let T be a triangle in X , with vertices p , q and r . A comparison triangle T ∗ in M k 2 for T is a triangle in M k 2 with vertices p ′ , q ′ and r ′ such that d ( p , q ) = d ( p ′ , q ′ ) , d ( p , r ) = d ( p ′ , r ′ ) and d ( r , q ) = d ( r ′ , q ′ ) .
Such a triangle is unique up to isometry.
The interior angle of T ∗ at p ′ is called the comparison angle between q and r at p . This is well-defined provided q and r are both distinct from p .
