In the limit as you keep stepping back, all possible viewing directions are parallel to the axis of the hyperboloid. Then what you see is exactly the entire, Euclidean xy-plane with the geodesics represented as single branches of any hyperbola whose center is at the origin (and also, in a limiting case, any straight line through the origin). This is just the (one sheet of a two-sheeted) hyperboloid model after it has been projected perpendicularly onto the xy-plane. (Assuming the z-axis is the axis of the hyperboloid.) Fortunately, you didn't ask about distances, since I'm feeling too lazy to compute what they would be on this model. (But the obvious one-to-one correspondence between the hyperboloid model and this xy-plane model can be used to transfer distances from the hyperboloid model to the corresponding points of the xy-plane model, thereby making the two isometric, as well they should be.) --Dan
On Jan 14, 2015, at 4:08 PM, Mike Stay <metaweta@gmail.com> wrote:
Suppose we have the hyperboloid t^2 - x^2 - y^2 = 1.
We get the Beltrami-Klein disk model of the plane by placing our eye at the origin and tracing what we see on the plane t=1. In this model, geodesics are straight lines, but angles aren't preserved.
We get the Poincare disk model of the plane by taking one step back, placing our eye at t=-1 and tracing what we see on the plane t=0. In this model, geodesics are arcs of circles intersecting the boundary of the disk at right angles and angles are preserved.
Clearly we can continue stepping backwards; what happens if we place our eye at t=-2 and draw what we see on the plane t=-1? Does it have any nice properties?
Is there a limiting model? Does it have any nice properties?
Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com