Figure 4: Hyperbolic reflections. (a) If the hyperbolic line is a diameter of γ, the hyperbolic reflection is simply the Euclidean reflection about the diameter. (b) If the hyperbolic line is a circular arc orthogonal to γ, then the hyperbolic reflection is an inversion about the hyperbolic line.
Theorem 5: Hyperbolic reflections are isometries. If points A ’, B’ are reflections of points A, B about a hyperbolic line, then .
  1. Hyperbolic line tool
Although in theory the defining circle γ can be any circle, for purposes of creating custom tools in GeoGebra for constructing hyperbolic lines and circles, we will take γ to be the unit circle. When constructing a hyperbolic line passing through two points, we must consider several possibilities depending on whether or not the points lie on a diameter of γ and whether the points are located in the interior or on the boundary of γ. These possibilities can be divided into three cases.