Theorem 1
Let p denote stereographic projection with vertex N
from


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,1t) for some real t.
Thus the points on C project to points on the curve 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. 
Stereographic projection 