Define ${\displaystyle M_{k}^{2}}$ as the 2-dimensional metric space of constant curvature ${\displaystyle k}$. So, for example, ${\displaystyle M_{0}^{2}}$ is the Euclidean plane, ${\displaystyle M_{1}^{2}}$ is the surface of the unit sphere, and ${\displaystyle M_{-1}^{2}}$ is the hyperbolic plane.

Let ${\displaystyle X}$ be a metric space. Let ${\displaystyle T}$ be a triangle in ${\displaystyle X}$, with vertices ${\displaystyle p}$, ${\displaystyle q}$ and ${\displaystyle r}$. A comparison triangle ${\displaystyle T*}$ in ${\displaystyle M_{k}^{2}}$ for ${\displaystyle T}$ is a triangle in ${\displaystyle M_{k}^{2}}$ with vertices ${\displaystyle p'}$, ${\displaystyle q'}$ and ${\displaystyle r'}$ such that ${\displaystyle d(p,q)=d(p',q')}$, ${\displaystyle d(p,r)=d(p',r')}$ and ${\displaystyle d(r,q)=d(r',q')}$.

Such a triangle is unique up to isometry.

The interior angle of ${\displaystyle T*}$ at ${\displaystyle p'}$ is called the comparison angle between ${\displaystyle q}$ and ${\displaystyle r}$ at ${\displaystyle p}$. This is well-defined provided ${\displaystyle q}$ and ${\displaystyle r}$ are both distinct from ${\displaystyle p}$.

References

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