Comparison Triangle

Define $M_^$ as the 2-dimensional metric space of constant curvature $k$. So, for example, $M_^$ is the Euclidean plane, $M_^$ is the surface of the unit sphere, and $M_^$ 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 $M_^$ for $T$ is a triangle in $M_^$ 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'',

Metric geometry

{{metric-geometry-stub