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