Hyperbolic arc of circumference

The tool that we have created allows to construct the hyperbolic arc of a circumference that goes through three determinated points. The order in which the points are marked is important since in the construction we will consider that the first point that the user marks is the origin of the arc; the second, an intermediate point and the third, the final point. Like in the case of the circumference that goes through three points the construction of the arc will not always be possible. It will be possible if and only if we can construct the circumference. Given three ordered points we construct the hyperbolic arc by the following steps:
  1. Plot the hyperbolic circumference that goes through the three given points.
  2. Plot the hyperbolic ray that begins in the center of the circumference and passes through the second given point.
  3. Consider the intersection of the ray plotted in the second step with the hyperbolic circumference.
  4. Plot the Euclidean arc with origin in the first given point, which passes through the intersection of the third step and finishes in the third given point.

We need the third step to affirm that if the hyperbolic circumference that passes through the three points does not exist the tool does not draw the arc, either. If the circumference exists there will be the intersection and this will be the second given point.

List of tools
Hyperbolic geometry