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I am interested in the shape that the following expression draws on the x-y plane as you vary $\psi$ where $z_x$ and $z_y$ are the x and y components of a complex 2D vector.

$x=\text{Real}(e^{i\psi}z_x),y=\text{Real}(e^{i\psi}z_y)$

In general it will be a rotated ellipse centered at origin. What is the best way of finding the orientation of the major axis and the minor axis?

Gappy Hilmore
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    Please clarify what $\text{Real}(z_1,z_2)$ means in this context. – dxiv May 07 '21 at 21:53
  • @dxiv $x=\text{Real}(e^{i\psi}z_x),y=\text{Real}(e^{i\psi}z_y)$ It is the real part of a a complex 2D vector. Real part of the first component is interpreted as the x coordinate, real part of the second component is interpreted as the y coordinate – Gappy Hilmore May 07 '21 at 23:44
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    Let $z_x=a+ib, z_y=c+id$, then $x=a\cos\psi-b\sin\psi$ and $y = c \cos\psi-d\sin\psi$. Solve the equations for $\cos\psi, \sin\psi$ in terms of $a,b,c,d,x,y$, then $\cos^2\psi + \sin^2\psi = 1$ is the equation of an ellipse in $(x,y)$. – dxiv May 08 '21 at 00:34
  • @dxiv given Ax^2+Bxy+Cy^2=1, how do I find the properties of the major axis? – Gappy Hilmore May 08 '21 at 01:46
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    See for example 1, 2, 3. – dxiv May 08 '21 at 01:55
  • @dxiv this works, thanks – Gappy Hilmore May 08 '21 at 05:48

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