hypercycles through a given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity.

Question

On http://en.wikipedia.org/wiki/Hypercycle_%28geometry%29 I found the statement.

The hypercycles through a given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity.

But I don't understand it.

hypercycles are curves (equidistant to some line)

• How do you get their tangent at a (given) point?

And what does the rest of the statement mean? can somebody give a proof and a picture?

Answer 1

Here is an illustration of what's going on.

Here I chose the common tangent to be orthogonal to the diameter through the given point. This ensures that the point lies on the symmetry axis of the crescent, so this is the most symmetric situation. For other tangents, the points where the hypercircle touches the boundary of the model would move at different speeds.