Comparison triangle

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.

References

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