In order to
measure the length of hyperbolic segments we need to create a
tool that allows us to measure
distances. Thus, with this tool we will
know the length of a hyperbolic segment. In order to use it,
it is only necessary to mark two points, which will be the endpoints of
The steps which are necessary to follow for measuring the distance are
based with the formula of the double ratio and are the following ones:
The double ratio for
four points is given by:
- Construct the hyperbolic
- Plot the Euclidean segments that we need to
apply the formula.
That is, we construct the hyperbolic line
that contains the
segment and consider the two intersection points with the
boundary line. Then, we plot the Euclidean segments that join these
intersections with the endpoints of the segment.
- Measure the Euclidean length of the four
segments that we have constructed in the former step.
- Apply the formula of the double ratio.
(u, v, s,
t) = (u-s)(v-t):(u-t)(v-s).
suppose the segment does not belong to one perpendicular
line to the boundary line we will always be able to make these steps.
List of tools