Hyperbolic line

The hyperbolic lines, in the Poincaré's Half-Plane Model, are the semicircumferences centered at a point of the boundary line and arbitrary radius and the euclidian lines perpendicular to the boundary line. In this model these two objects are considered as lines so that the concept of geodesics (the curve that minimizes the distance between two points) still remain true.

The hyperbolic line tool we have constructed only allows us to draw the hyperbolic lines that are semicircumferences. This restriction is because, in the construction we use an intersection that does not exist when the hyperbolic line is also an euclidean line.

Let's see how we have constructed the tool when you have two points in the allowed position:

- Draw the Euclidean segment ending in the two given points.
- Construct the midpoint.
- Draw the perpendicular line that contains the segment of (1) and that passes through the midpoint.
- Consider the
intersection of the constructed perpendicular line and the boundary
line. This point of intersection will be the center of the
circumference, which will give us the hyperbolic straight line. (This
is the intersection that does not exist when the two given points are
in the same perpendicular line.)

- Construct the circumference with center at the last intersection and that passes through one of the two given points.
- Mark the intersection of this circumference with the boundary line.
- Plot the arc of circumference with endpoints in the intersections of (7) and passes through one of the given points. This semicircumference is the hyperbolic line that goes through the two fixed points.

Some steps in the
construction of a hyperbolic line.

This construction allows us to find the hyperbolic line. In the steps (2) and (3) we construct the geometric place of the points of the plane that are equidistant of the two points given, that is, the perpendicular bisector. Therefore, the center of the circumference that we search is on this line but also has to be on the boundary line, by definition of hyperbolic straight line. Thus, on step (4), the center of the circumference is determinated.

To get that the sketch only shows the semicircumference we hide, with the hidden tool, all other objects that we have constructed.

List of tools

Hyperbolic geometry