Let p denote stereographic projection with vertex N
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.
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.