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

• M Bridson & A Haefliger - Metric Spaces Of Non-Positive Curvature, ISBN 3-540-64324-9