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.