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’.