Figure 1: Lines in the Poincaré disk model are either open diameters or open circular arcs orthogonal to γ. Each line is shown passing through points A and B and with their ideal endpoints P and Q.
Circles orthogonal to γ can be constructed in several different ways. One method is to use the following result from Euclidean geometry which makes use of tangents [11].
Definition 1: Two circles are orthogonal if their tangent lines are perpendicular at the point of intersection.
Theorem 1: Let c be a circle with center O and c’ be a circle with center O’ and suppose the circles intersect at points A and B. Then the circles are orthogonal if and only if the tangent lines to c’ at A and B intersect at O and the tangent lines to c at A and B intersect at O’.