 Hyperbolic ray

To construct a hyperbolic ray we need two ordered points. The order we fix the two points is important since we will consider that the hyperbolic ray has the origin in the first given point.

Because of that, to construct a hyperbolic ray with the Sketchpad we have to have in account that the order with which the points are marked will be the order that we will use, that is, the first marked point will be the origin of the ray.

Let's see the steps followed to construct a ray:
1. Draw the circumference that passes through the two given points and has the center on the boundary line repeating the five first steps followed to construct the hyperbolic line.
2. Draw the Euclidean line through the first point and the center of the circumference.
3. Consider the intersection of this line and the circumference. There are two points of intersection since the line we have constructed is a prolongation of a diameter of the circumference. One intersection is the first given point. We will need the other intersection point.
4. Construct the arc of circumference that has origin in the first given point, goes through the second given point and finishes in the intersection of the former step.
5. Consider the intersection of the arc of the former step with the boundary line. This point, lying in the boundary line, will give us where the hyperbolic ray finishes.
6. Draw the arc of circumference that joins the first given point, passes through the second and finishes in the intersection point considered at (5). This arc of circumference is the hyperbolic ray we wanted to construct.  If we prove that step (4) is well defined in any situation we will have proved that this construction is correct. This is true because the point we suppose that will always be the third is the antipodal point of the first given point and, to be the antipodal, will be on the other half of the plane determined by the boundary line. The two given points will always be on the same half of the plane since they are hyperbolic points. So, we will always be able to make this construction and we will always obtain the ray in the correct direction. Once made the construction it is only necessary to hide all objects except the last arc of circumference.