You can also do this:
S3 = {(z, w) ∈ C2 | zz* + ww* = 1}
S2 = C U {∞}
f: (z, w) ↦ z/w
S1((z, w)) = {(zeiθ, weiθ) | θ ∈ [0, 2π)}
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