 Hyperbolic segment

To construct the hyperbolic segment, as in the former cases, we need two points. In this case, the order will not be relevant. The steps we will follow will be essentially the same ones that we used in the hyperbolic line tool, since a hyperbolic segment belongs to a hyperbolic line.

1. Repeat the steps followed to construct the hyperbolic line. Do not hide any object.
2. Consider the intersection of the Euclidean perpendicular line to the Euclidean segment that joins the two given points with the semicircumference that gives us the hyperbolic line. This two objects have been constructed in step (1).
3. Draw the arc of circumference that passes through one of the given points, the intersection we have marked in the former step and the other point, in this order. This is the hyperbolic segment that we search.  The hyperbolic segment is well constructed because the intersection of the second step will always exist and it will be between the two given points. The perpendicular line considerated in the first step is the perpendicular bisector of the Euclidean segment. So, it passes through the center of the circumference, then we know that it has to intersect it in two points diametrally opposite. Thus, we know that the intersection always exists and that the step (2) is well defined.

It is also necessary that the intersection point that belongs in the same half-plane that the given points will be between these two given points. That is, if we follow the arc in the order that indicates the step (3) we find the points in the indicated order. Suppose that it is not true. In general, every line divides the plane in two parts. We can draw the lines that joins the center with one of each given points. If our assumption is not true one of these two lines will separate the intersection point and the other given point. But it is not possible because the intersection point is at the same distance from the two given points and in this situation it is not possible. Thus, we will be able to make this construction whenever the given points do not belong to the same line perpendicular to the boundary line. Finally, we only have to hide all the constructed objects except the last arc.

List of tools
Hyperbolic geometry