Stereographic Projection

Complex Analysis · Axiom Academy

EXAMPLE Stereographic Projection Sending a complex number to its point on the Riemann sphere We use the unit sphere centred at the origin, with north pole N=(0,0,1) (which represents ) and south pole S=(0,0,-1) (which represents 0 ). The stereographic projection of z=x+iy onto the sphere is Find the image P(z) of z=1+i , confirm it lies on the unit sphere, and recover z from that point. The line from the north pole N through a sphere point hits the plane at z . Projection reverses it: z on the plane lifts to P(z) on the sphere. Nice work. You projected a complex number onto the Riemann sphere, checked the image really sits on the sphere, and ran the map backwards. The formula: , so for z=1+i the image is . Always on the sphere: X^2+Y^2+Z^2=1 for every z — the third coordinate measures height. Landmarks: ; the unit circle |z|=1 maps to the equator Z=0 ; and drives , so N stands for . Reversible: from a sphere point (X,Y,Z) with , recover . This one-to-one correspondence is exactly what lets us treat as an ordinary point and view as a sphere.

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