This custom tool can be used both as a tool and as the command Pline[
<Point>, <Point> ] in
the Input bar. In addition, the custom tool can be saved to your
computer as GGT file and then imported into other GeoGebra files. To do
this, select Tools/Manage Tools, then select Save As and press Save.
Then open a new GeoGebra file and use File/Open to import the GGT file.
Be sure to do this prior to beginning the hyperbolic circle
construction, since this approach hides all of the details of
construction of the hyperbolic line tool and allows you to focus on
constructing the hyperbolic circle. This works since all of the details
of the construction are contained in the GGT file and they are not
displayed in the Algebra window.
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Hyperbolic circle tool
Hyperbolic circles in the Poincaré disk model are also Euclidean circles
but in general the hyperbolic center is not the same as the Euclidean
center. The key to constructing a hyperbolic circle with center at
points A and passing through point B is to locate its
Euclidean center C and then construct a Euclidean circle with
center C passing through B. The construction of the
hyperbolic circle is broken into three cases depending on whether or not
A = O and whether or not the A and B are
collinear with O.
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Hyperbolic center at the origin
If A = O, then by Theorem 4 the Euclidean center of the
circle is also at O and the hyperbolic circle can be constructed
using a Euclidean circle centered at O and passing through
B . Begin the construction by creating γ and two points inside γ.