Angle of parallelism

The angle of parallelism has an important role in the Hyperbolic Geometry since it relates lengths and angles. In Euclidean Geometry it is not possible to find any similar relation.

To describe the angle of parallelism we need to fix a line, r, and an exterior point P. Then, we consider the perpendicular line to r that passes through P and its intersection, Q, with r. The length of the segment PQ coincides with the distance from P to r. Now, we consider the two parallel lines to r that contains P and the angle between one of these parallel lines and the perpendicular line. This angle is defined as the angle of parallelism.

We have two tools to calculate the angle of parallelism. One of them only calculates the angle of parallelism without making any apparent modification in the construction and the other one, apart from calculating its value, represents this angle and draws the two parallel lines. For both cases we need three ordered points, so that the two first belong to the line. Now, we calculate the value of the angle of parallelism:

  1. Construct the hyperbolic line that joins the two first points.
  2. Plot the parallel lines.

  3. Now, we calculate the value of the angle of parallelism from the Euclidean angle since in this model the hyperbolic angles coincide with the Euclidean ones.
  4. Plot the tangent line to the two Euclidean circumferences that define each one of the parallel line containing the outer point.
  5. Calculate the Euclidean angle between the two lines in (3). This angle measure twice the angle of parallelism.
  6. Divide the last measure by two. This is the value of the angle of parallelism.

We hide all objects except from the result of the step (5). If we want to draw the bisector it is necessary to
go on with the following steps and not to hide any of the former objects.
  1. Plot the two hyperbolic segments that have as an endpoints the outer point and the intersection point of the line given with the boundary line. In fact, these two hyperbolic segments that we have constructed are not hyperbolic segments (they have a point in the boundary line) but we want these segments to plot the arc of circumference that joins both points, so we can use this construction.
  2. Plot the hyperbolic angle bisector of the angle between the two former segments. Observe that we can think each segment as a hyperbolic ray.
  3. Consider the intersection point of the hyperbolic angle bisector with the line in (1).
  4. We plot the hyperbolic segment that joins the two former points. This is the segment that defines the angle of parallelism.
Observe that the step (9) cannot be made in any situation. If the hyperbolic segment is also a Euclidean segment we cannot construct it with the hyperbolic segment tool. If it is the case, drag some of the initial points so that in the new position the segment could be constructed.

List of tools
Hyperbolic geometry