Figure 8: Construction of the hyperbolic circle c2 with
hyperbolic center A ≠ O and passing through B for
the case in which points A, B and O are not
collinear. All hyperbolic lines passing through the center A are
orthogonal to the circle. Since line OA is one such line, it is
orthogonal to c2 and the center of c2 must lie on this
line (if a line cuts a circle orthogonally, the line must be a diameter
of the circle). In addition, c2 is orthogonal to the hyperbolic
line pAB (circular arc) passing through points A and
B . By Theorem 1, the Euclidean center O2 of c2 must
lie on the tangent tB to the hyperbolic line pAB at point
B . Thus the Euclidean center O2 of c2 is located at
the intersection of line OA and tangent tB. The hyperbolic
circle c2 can then be constructed using a Euclidean circle
centered at O2 and passing through B.
Begin by hiding c1 from the previous case and then implement the
construction using the following GeoGebra code. Although the most
obvious way to construct the tangent is to use the Tangent[
<Point>, <Conic> ]
command, we forego this approach since the command fails for some
locations of B on γ (returns undefined, most likely since the
numerical algorithms occasionally fail to find the tangent at the
endpoint of an arc). The approach taken here is to use the fact that the
tangent to a circle is perpendicular to its radius.