Hyperbolic circumference: three points

We can also plot the circumference that goes through three given points. This construction will not always be possible to be done since not always, given three points, there is a hyperbolic circumference that contains them, as we proved in Workshop of Hyperbolic Geometry.pdf (in Catalan). Anyway, if it exists it is unique. To construct it, it is necessary to fix three points of the half-plane model. The steps that we have followed are the following ones:

- Plot the hyperbolic triangle that joins the three given points. We can construct it plotting the three hyperbolic segments. Anyway, we have created a tool that allows us to construct the hyperbolic triangle given three points.
- Plot the hyperbolic perpendicular bisector for two of the three segments.
- Consider the intersection of these two.
- Plot the hyperbolic circumference with center in the former intersection point and goes through one of the three given points. This is the hyperbolic circumference which we want to construct.

This construction, as we have commented, will not always be possible. The step that will fail will be the third. The perpendicular bisectors will not cut in the half-plane. In this case, no circumference will be constructed since in the sketch of the hyperbolic perpendicular bisector we only draw the semicircumference that belongs to the half-plane. So, the construction will detect correctly when it is not possible to construct the hyperbolic circumference that goes through three given points.

List of tools

Hyperbolic geometry