Look at the two planes that are determined by the North Pole N and vectors tangent to the great circles at X . Notice that the intersection of these two planes with the horizontal image plane determines the image of the angle. In particular, consider the plane determined by X and the North and South Poles, the plane tangent to the sphere at X , and the planes tangent to the sphere at the North and South Poles. Determine the relationships among these planes. Introduce a coordinate system and find a formula for the function s -1 from the plane to the sphere which is the inverse of the stereographic projection s .
Stereographic projection is circle-preserving.
See Figure 16.4. Thus the segments BP and AP are tangent to the sphere and have the same length r . Now we will study other models of the hyperbolic plane. Let D 2 be the disk model of a hyperbolic plane and assume its radius is 2. Then place a sphere of radius 1 tangent to the disk at its center.