Figure 4: Hyperbolic reflections. (a) If the hyperbolic line is
a diameter of γ, the hyperbolic reflection is simply the Euclidean
reflection about the diameter. (b) If the hyperbolic line is a circular
arc orthogonal to γ, then the hyperbolic reflection is an inversion
about the hyperbolic line.
Theorem 5: Hyperbolic reflections are isometries. If points
A ’, B’ are reflections of points A, B about
a hyperbolic line, then .
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Hyperbolic line tool
Although in theory the defining circle γ can be any circle, for purposes
of creating custom tools in GeoGebra for constructing hyperbolic lines
and circles, we will take γ to be the unit circle. When constructing a
hyperbolic line passing through two points, we must consider several
possibilities depending on whether or not the points lie on a diameter
of γ and whether the points are located in the interior or on the
boundary of γ. These possibilities can be divided into three cases.