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Math notes (quick)
We sample base points n on S^2 and use a standard Hopf section:
z1 = sqrt((1+n_z)/2), z2 = (n_x + i n_y)/sqrt(2(1+n_z)). The fiber over n is
( e^(i·theta) · z1 , e^(i·theta) · z2 ) ⊂ S^3. We embed S^3 → R^3 by stereographic projection from (-1,0,0,0):
(x0,x1,x2,x3) → (x1, x2, x3)/(1 - x0). Varying theta draws each circular fiber. Colors encode base directions.