Figure 9: Construction of the hyperbolic circle c3 with
hyperbolic center A and passing through B for the case in
which A ≠ O and A and B are collinear with
O . Begin by determining the inverse of A with respect to γ
and then construct circle ca passing through A and
A ’ and centered at the midpoint of segment AA’. By Theorem
2, ca is orthogonal to γ and is therefore a hyperbolic line in
the Poincaré disk. Define A” and B” as the inverses of
A and B with respect to ca (be careful since in all
previous constructions, the inverses have been with respect to γ). To
finish the construction, we use the distance metric. By Theorem 5,
d (A”, B”) = d(A, B).
Furthermore since A is invariant after inversion with respect to
ca , we obtain d(A, B”) = d(A,
B ). Hence, the hyperbolic circle c3 can be constructed
using a Euclidean circle centered at the midpoint of segment BB’’
and passing through the point B.
Begin by hiding all of the objects from the previous construction except
points A and B and the defining circle γ and then move the
A and B onto a diameter. Use the following GeoGebra code
to implement the construction.