§ Complex orthogonality in terms of projective geometry

If we think of complex vectors p=[p1,p2]p = [p_1, p_2], q=[q1,q2]q = [q_1, q_2] as belonging to projective space: that is, pp1/p2p \simeq p_1/p_2, and qq1/q2q \simeq q_1 / q_2, we can interpret orthogonality as:
p.q=0p1q1+p2q2=0p1/p2=q2/q1p=1/q=q/q \begin{aligned} p . q = 0 \\ p_1 \overline q_1 + p_2 \overline q_2 = 0 \\ p_1 / p_2 = - \overline{q_2} / \overline{q_1} \\ p = -1/\overline{q} = -q/|q| \\ \end{aligned}
If we imagine these as points on the Riemann sphere, TODO

§ References

  • Visual Complex analysis by Tristan Needham