Figure 3: A hyperbolic circle in the Poincaré disk is also a
Euclidean circle. The hyperbolic center is offset towards the boundary
of the Poincaré disk since distances near the boundary are larger.
An isometry is a distance preserving transformation. There are four
isometries in the hyperbolic plane including hyperbolic reflections,
hyperbolic translations, hyperbolic translations and the parabolic
isometry which has no Euclidean counterpart. In this paper, we focus on
the hyperbolic reflections [8] since they are required for the
construction of the hyperbolic circle.
Definition 5: A hyperbolic reflection is either a Euclidean
reflection across a diameter or an inversion with respect to a circular
arc orthogonal to γ.