How does the angle of parallelism relate to the arc of a circle and a point outside?
In Hyperbolic Geometry, I'm trying to figure out what happens to the "visibility" of a circle when a point outside increases in distance away from the circle.
So given a point p outside the circle, there exists two tangents from the circle that cross point p. As p moves farther away these tangents move and the arc between them increases. I'm given the equations for the angle of parallelism(2arcTan(e^-x)) and the length of the circle (2piSinH(r)).
I know the angle goes to 0 as x->infinity.
I would prefer the answer to finding an expression to finding the "visibility" of the circle arc or at least a hint in the right direction to be able to know where to assume the angle of parallelism lies in relation to a point outside a circle.