This lab is to help you write a Euclidean script that constructs the hyperbolic line (an arc of a Euclidean circle orthogonal to the boundary circle for the disk model of the hyperbolic plane).

Your script will have four points as givens; two points are the points the line is to pass through, and the other two are the center of the disk and a point on the boundary circle of the hyperbolic plane.

When you're done, and you have a situation like:

You'll be able to play your script, choosing A, B, P, and Q in the order appropriate to how you constructed the script, and produce:

And, you'll be able to drag P or Q around to see what the line looks like when the two points are in different positions.

So, how to go about making the construction? To locate the center of the required arc, recall two ideas:

A. The Chord Theorem says that the center must be on the perpendicular bisector of the chord PQ

B. The Lab 13 investigation showed that any circle orthogonal to a given circle and passing through a given point would also have to pass through the inverse of the given point in the given circle.

Using the above two ideas, you can make the construction of the center of the required arc. Once you have the center, the rest is easy.

Hints:

1. Finding the center

2. Making the arc for the hyperbolic line

3. Making the arc for the hyperbolic line segment