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