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Stereographic projection on complex plane

Let V be a circle lying in S. Then there is a unique plane P in R^3 such that p / S = V ( / = intersection). Recall from analytuc geomerty that
P = { (x_1,x_2,x_3) : x_1 b_1 + x_2 b_2 + x_3 b_3 = L, where L is a real number}.
Where ( b_1,b_2,b_3) is a vector orthogonal to P . It can be assumed that (b_1)^2 + (b_2)^2 + (b_3)^2=1. Use this information to show that if V contains the point N then its seteographic projection on the complex plane is a straight line. Otherwise, V projects onto a circle in complex plane.

N = (0,0,1) the north pole on S
Please explain to me every step, I don't just want the answer, but I also want to understand it and be able to work similar problems.

Solution Preview

Proof:

V is a circle. V determines a unique plane P in R^3 and V lies in this plane. So P^S=V.
We consider two cases.
Case 1: V contains N=(0,0,1). Since V=P^S, then N is also on the plane P. The equation of P is x_1*b_1+x_2*b_2+x_3*b_3=L, where (b_1)^2+(b_2)^2+(b_3)^2=1. We plug in (0,0,1) and get b_3=L. We also know that the center of the sphere S is (0,0,0) and this center is also the center of V. So we have L=0. ...

Solution Summary

A stereographic projection on complex plane is investigated.

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