Question:
Prove algebraically that the stereographic projection of a circle
Prove algebraically that the stereographic projection of a circle (C) lying in a sphere (S) is either a circle or a straight line.
Definition:
Stereographic projection is done from the north pole of the sphere onto a plane tangent to the sphere at its south pole.
Hint:
A circle on the sphere is contained in a plane (P), so that C = P/S. A plane can be defined by equation P = {(x, y, z): a x + b y + c z = d, where a, b, c, and d are constants and (x, y, z) are Cartesian coordinates in R^3
The hint suggests an algebraic proof (those interested in a geometric proof may find one by Yana Zilberberg Mohany at https://math.ucsd.edu/~mohanty/nopix1.html#Eves2).