Comparison triangle

From Wikipedia, the free encyclopedia

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 $p$ .