Centres of a Circle?
Draw a Euclidean circle inside the unit disc and note the coordinates of
its centre and its radius. We know this is also a non-Euclidean (hyperbolic)
circle but we wish to calculate the coordinates of its non Euclidean centre
C'. To do this we can use the Cabri calculator together with the formulas
for exchanging distances in the two geometries. In this model the two centres
will lie on a line through the centre O so we only need use tanh and arctanh.
Now A and B lie on a diameter of both circles and we can calculate the
non Euclidean distances OA and OB given the information we have about their
Euclidean distances. We now know the nonE diameter and nonE radius and
so can determine the nonE distance OC'. This can finally be put into
Euclidean distance terms and coordinates.
Hyperbolic Circles
Other curves; horocycles
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