Hyperbolic
circumference: center and point
To make the
construction we will suppose that the point does not belong to the
perpendicular line to the boundary line. We also suppose that the first
given point is the center and the
second the point from which we want the circumference to pass.
The steps to be done are the following ones:
- Plot the perpendicular line to the boundary line that goes
through the center, C, (the
first point).
- Plot the circumference with center in the boundary line and
that
goes through C and D. We will always be able to plot
this circumference since we suppose that C and D do not belong in the same
perpendicular line to the boundary line, where it is the center of this
circumference.
- Plot the Euclidean line that passes through one
of the intersection points of the former circumference with the
boundary line and for D.
- Consider the intersection between the line plotted in the
first step and the line plotted in the third step. We name B this intersection. It can be
proved (Workshop
of Hyperbolic Geometry.pdf, in
Catalan) that this intersection exists and it is a point of
the hyperbolic circumference that we want to determine.
- To construct the other point of
the hyperbolic circumference that belongs to the line plotted in the
first step, we used the homothety with center in the foot of
the perpendicular in first step, O
and reason CO·CO/BO. B is transformated into a
point E that it is at the same hyperbolic distance from the
center than the point D and B to be homotheties isometries for
the half-plane model.
- Construct the Euclidean midpoint,
M, of the Euclidean
segment ending in B and C.
- Plot the Euclidean circumference with center M and radius the segment
from M to D. This circumference is the
hyperbolic
circumference we wanted.
To construct the
hyperbolic circumference we used that the homotheties are isometries
for
the half-plane model, so they preserve distances. Moreover, in Workshop
of Hyperbolic Geometry.pdf (in Catalan) we
proved that the third axiom is true using almost the same construction
we have followed here. The only difference is that there we have the
point in the same perpendicular to the boundary line as the center and
here we suppose that it is not true. So, the first thing we did is to
transport the point in the same
perpendicular, keeping the distance
to the center. Then, we are in the
same hypothesis.
List of tools
Hyperbolic
geometry
