Hyperbolic Horocycle

In the Half-Plane model of the Hyperbolic Geometry, a horocycle is either an Euclidean line parallel to the boundary line or a circumference tangent to the boundary line. This construction will give us both horocycles, that passes through a fixed point. To make the construction we will think a horocycle as the the limit of a hyperbolic circumference when its radius tends to infinity. Then, we will need two points, one will be the hyperbolic center of the circumference and the other the point in the horocycle.

- Plot the hyperbolic circumference that has center in the first given point and passes through the second.
- Plot the Euclidean parallel line to the boundary line that passes through the second point. This line is one of the horocycles.
- Plot the perpendicular to the boundary line that passes through the center of the hyperbolic circumference in (1).
- Construct the intersection between the perpendicular line in
(3) with the boundary line.

- Plot the Euclidean segment with endpoints in the second fixed and in the intersection in (4).
- Plot its Euclidean midpoint.
- Construct the Euclidean perpendicular line to the segment that passes through the midpoint. This line is the Euclidean perpendicular bisector.
- Construct the intersection between the perpendicular bisector and the perpendicular line in (3).
- Plot the Euclidean circumference with center in intersection (8) and passes through the second fixed point. This is the other horocycle.

In the following steps we will plot the Euclidean perpendicular bisector to the segment determined by the second fixed point and the intersection in (4). To trace the perpendicular bisector we will follow the same steps we made in the constructions we need it:

Observe we can see what we have named as horocycles fulfill the definition. If we drag the center of the hyperbolic circumference for the perpendicular in (3) towards the boundary line we will obtain the second constructed horocycle and if we drag it in the other direction we will obtain the first constructed horocycle. However, notice that if we reproduce this construction the circumference tangent to the boundary line will not be plotted. This is because, it is used the circumference given the center and a point tool and this tool had been constructed to plot a circumference if and only if all of its points are hyperbolic.

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Hyperbolic geometry