Hyperbolic length

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 segment. 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:
  1. Construct the hyperbolic segment.
  2. 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.
  3. Measure the Euclidean length of the four segments that we have constructed in the former step.
  4. Apply the formula of the double ratio.
The double ratio for four points is given by:

(u, v, s, t) = (u-s)(v-t):(u-t)(v-s).
If we 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
Hyperbolic geometry