Hyperbolic translation
A hyperbolic translation is a transformation of the hyperbolic plane
which
has only one fix point at the boundary. A hyperbolic rotation is the
composition of two
reflections which intersect assimptotically, at the fix point.
Anyway, in this macro we also consider the translations which have as a
fix point the point which does not lie in the boundary line. To find
the translated point of a given point we have just to fix the distance
and the direction. Then, we use the euclidean translation along the
euclidean line parallel to the boundary line (which is a horocycle) and
contains the point we want to transform.
The macro
we have created allows also to translate segments,
triangles and circles. To construct this other tools we just transform
each of the defining points and then we construct again the object from
the images points.
Note that
the translation of a horocycle which is an euclidean line parallel to
the boundary line are invariants.
In
the next figure it
is plotted the translation of a triangle.
Hyperbolic geometry
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