a comparison triangle

Suppose you have a triangle drawn on some curved or singular space, and you want to ask 'is this space more curved than flat, or less?' You cannot read curvature off the bare space directly. The trick is to build a stand-in: take the very same three side lengths and lay out a triangle with those lengths in a known model surface — the flat plane, a sphere, or the hyperbolic plane — then compare. The triangle you lay out in the model is the comparison triangle, and comparing the real triangle to it is how curvature gets defined in spaces with no derivatives.

Concretely, fix a model curvature kappa and let M_kappa be the model surface of constant curvature kappa: the sphere of radius 1/sqrt(kappa) when kappa > 0, the Euclidean plane when kappa = 0, the hyperbolic plane scaled by 1/sqrt(-kappa) when kappa < 0. Given a geodesic triangle in your space X with vertices p, q, r and side lengths a = d(q,r), b = d(p,r), c = d(p,q), the comparison triangle is the triangle p', q', r' in M_kappa with the same side lengths (it exists and is unique up to congruence, provided the perimeter is small enough when kappa > 0). For any point x on a side of the real triangle there is a corresponding comparison point x' on the matching side of the model triangle, at the same distance from the endpoints. The whole theory then compares the real distance d(x, y) between two such points to the model distance |x' - y'|.

Comparison triangles are the single device that lets curvature bounds be stated without calculus. 'Curvature bounded above by kappa' (CAT(kappa)) means real triangles are thinner than their comparison triangles: d(x, y) <= |x' - y'|. 'Curvature bounded below by kappa' (Alexandrov) means real triangles are fatter: d(x, y) >= |x' - y'|. The very same comparison provides Toponogov's theorem in the smooth Riemannian world, which is why this metric definition is the right generalization. One caution: when kappa > 0 the comparison triangle only exists if the perimeter is below 2pi/sqrt(kappa) (the perimeter of the model space), so the definitions restrict to triangles of bounded size.