stereographic projection on complex plane

[Meital]

Stereographic projection on complex plane

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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 any hints will be appreciated!!