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Define M k 2 {\displaystyle M_{k}^{2}} as the 2-dimensional metric space of constant curvature k {\displaystyle k}. So, for example, M 0 2 {\displaystyle M_{0}^{2}} is the Euclidean plane, M 1 2 {\displaystyle M_{1}^{2}} is the surface of the unit sphere, and M − 1 2 {\displaystyle M_{-1}^{2}} is the hyperbolic plane.
Let X {\displaystyle X} be a metric space. Let T {\displaystyle T} be a triangle in X {\displaystyle X} , with vertices p {\displaystyle p} , q {\displaystyle q} and r {\displaystyle r}. A comparison triangle T ∗ {\displaystyle T*} in M k 2 {\displaystyle M_{k}^{2}} for T {\displaystyle T} is a triangle in M k 2 {\displaystyle M_{k}^{2}} with vertices p ′ {\displaystyle p'} , q ′ {\displaystyle q'} and r ′ {\displaystyle r'} such that d = d {\displaystyle d=d} , d = d {\displaystyle d=d} and d = d {\displaystyle d=d}.
Such a triangle is unique up to isometry.
The interior angle of T ∗ {\displaystyle T*} at p ′ {\displaystyle p'} is called the comparison angle between q {\displaystyle q} and r {\displaystyle r} at p {\displaystyle p}. This is well-defined provided q {\displaystyle q} and r {\displaystyle r} are both distinct from p {\displaystyle p}.