Exercise 2.4

Exercise 2.4.  Show that the projective representation of a hyperbolic straight line may be obtained from the conformal representation of a hyperbolic straight line by expansion from the center by a factor of M, and that:

Solution:

Above is a conformal pro jection on a minor circle c, with radius r , that we project onto a projective representation on the great circle C, which has radius R and origin O. Let r_c be the Euclidean distance from O to any point on the conformal repreesentation on C , and let b be the Euclidean distance from r_c to its corresponding point on the pro jective representation, so that the additive Euclidean distance from O to any point on the projective representation is b + r_c. Let M be the expansion factor from the Euclidean conformal distance to the Euclidean projective distance, so that:

                   (1)

Now, it is evident from the similarity of the two triangles that:

The problem is how to restate d in terms of R, r_c,  and b.  Given that the segment extending from O and joining d is equal to R, by the Pythagorean Theorem we know that 

So we rewrite the previous expression as 

And solve for b:

With this we can rewrite (1) and solve for M: