Conic in a complex plane

Question

This question may have been asked before..

what is the complex form through five points of a conic section in the complex plane?

Thanks for any hints.

Answer 1

In complex plane, we use $z$ and $\bar z$ as bases. Hence,

$$0=F(z,\bar z) \equiv \det \begin{pmatrix} (z+\bar z)^2 & z^2-\bar z^2 & (z-\bar z)^2 & z & \bar z & 1 \\ (z_1+\bar z_1)^2 & z_1^2-\bar z_1^2 & (z_1-\bar z_1)^2 & z_1 & \bar z_1 & 1 \\ (z_2+\bar z_2)^2 & z_2^2-\bar z_2^2 & (z_2-\bar z_2)^2 & z_2 & \bar z_2 & 1 \\ (z_3+\bar z_3)^2 & z_3^2-\bar z_3^2 & (z_3-\bar z_3)^2 & z_3 & \bar z_3 & 1 \\ (z_4+\bar z_4)^2 & z_4^2-\bar z_4^2 & (z_4-\bar z_4)^2 & z_4 & \bar z_4 & 1 \\ (z_5+\bar z_5)^2 & z_5^2-\bar z_5^2 & (z_5-\bar z_5)^2 & z_5 & \bar z_5 & 1 \\ \end{pmatrix}$$

and may further rearrange into

$$\bar u z^2+pz\bar z+u\bar z^2+\bar v z+v\bar z+q=0$$

with $p,q\in \mathbb{R}$.

Please see also my older posts in here and here.