Proof of Theorem 1

 Theorem 1 Let p denote stereographic projection with vertex N from the sphere S to the plane P. Then, for any circle C on S, p(C) is a line if N lies on C, and a circle otherwise. Proof We choose coordinates so that S is the sphere x²+y²+z² = 1 and N is the point (0,0,1) We take as P the plane z=0. Suppose that P on S has projection Q on P. Then Q = (X,Y,0) for some X and Y and P is the point where NQ cuts S. Suppose that C lies on the plane F: ax+by+cz = d. A point on NQ has the form (tX,tY,1-t) for some real t. This is on S if and only if (tX)²+(tY)²+(1-t)² = 1. Simplifying, we get t((X²+Y²+1)t-2)=0. Now, t=0 gives N, so P must have t=2/(X²+Y²+1). But P also lies on F, so that 2aX/(X²+Y²+1) +2bY/(X²+Y²+1) + c(1-2/(X²+Y²+1) = d. Simplifying, 2aX + 2bY - 2c = (d-c)(X²+Y²+1). Thus the points on C project to points on the curve 2ax + 2by - 2c = (d-c)(x²+y²+1). If c=d, this is a line, otherwise it is a circle. Finally, we note that c=d if and only if N = (0,0,1) lies on F.