The Ellipse

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Viewing the conic section $\phi$ maps out on the DC sector plane, we know that it follows an elliptical path centered on it's x axis. Performing a passive rotation on points in the DC section plane does not physically change the position in space, i.e. passive rotations only give the components in a new coordinate system. Once such a rotation has been performed, the equation describing these points must be done within that plane.

An ellipse centered at the origin can be expressed in the form

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

For an ellipse not centered on the origin, but instead the point (h',k'), this expression becomes

$\frac{(x+h')^2}{a^2}+\frac{(y+k')^2}{b^2}=1$

In the plane of the DC sector, this equation becomes

$\frac{(x'+\Delta a)^2}{a^2}+\frac{(y')^2}{b^2}=1$

where the center of the ellipse is found at $\{\Delta a, 0\}$.

Switching to the frame of the wires, the ellipse is still centered at $\{\Delta a,0\}$ in the DC sector, with the semi-major axis lying on the x' axis. For a rotation in the y-x plane, this corresponds to a positive angle $\theta$, with the rotation matrix $R(\theta_{yx})$. In the frame of the wires, this center point falls at

$\begin{bmatrix} x'' \\ y'' \\ z'' \end{bmatrix}= \begin{bmatrix} cos\ 6^{\circ} & -sin\ 6^{\circ} & 0 \\ sin\ 6^{\circ} &cos\ 6^{\circ} & 0 \\ 0 & 0 & 1 \end{bmatrix}\cdot \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}$

$\begin{bmatrix} x'' \\ y'' \\ z'' \end{bmatrix}= \begin{bmatrix} cos\ 6^{\circ} & -sin\ 6^{\circ} & 0 \\ sin\ 6^{\circ} &cos\ 6^{\circ} & 0 \\ 0 & 0 & 1 \end{bmatrix}\cdot \begin{bmatrix} \Delta a \\ 0 \\ 0 \end{bmatrix}$

$\begin{bmatrix} x'' \\ y'' \\ z'' \end{bmatrix}= \begin{bmatrix} \Delta a\ cos\ 6^{\circ} \\ \Delta a\ sin\ 6^{\circ} \\ 0 \end{bmatrix}$

$(x'', y'', z'')_{center} = (\Delta a\ cos\ 6^{\circ} , \Delta a\ sin\ 6^{\circ} , 0 )= (h'', k'', 0)$

Performing an active rotation, we will rotate the equation for an ellipse in the frame of the DC to the frame of the wires . In the frame of the DC, the ellipse is centered on the x' axis, with the intersection points not having a uniform spacing in the ellipse parameter. In the frame of the wires, the ellipse is tilted $6^{\circ}$ counterclockwise from the x axis, with the intersection points having a uniform spacing in the x component.

$\begin{bmatrix} x'' \\ y'' \\ z'' \end{bmatrix}= \begin{bmatrix} cos\ 6^{\circ} & sin\ 6^{\circ} & 0 \\ -sin\ 6^{\circ} &cos\ 6^{\circ} & 0 \\ 0 & 0 & 1 \end{bmatrix}\cdot \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}$

$\begin{bmatrix} x'' \\ y'' \\ z'' \end{bmatrix}= \begin{bmatrix} x'\ cos\ 6^{\circ}+y'\ sin\ 6^{\circ} \\ -x'\ sin\ 6^{\circ}+y'\ cos\ 6^{\circ} \\ 0 \end{bmatrix}$

Substituting this into the equation for an ellipse in the frame of the wires, that is parallel to the frame of reference x and y axis,

$\frac{(x''+h'')^2}{a^2}+\frac{(y''+k'')^2}{b^2}=1$

$\frac{((x'+\Delta a\ cos\ 6^{\circ})cos\ 6^{\circ}+(y'+\Delta a\ sin\ 6^{\circ})sin\ 6^{\circ})^2}{a^2}+\frac{(-(x'+\Delta a\ cos\ 6^{\circ})sin\ 6^{\circ}+(y'+\Delta a\ sin\ 6^{\circ})cos\ 6^{\circ})^2}{b^2}=1$

$\frac{(x'\ cos\ 6^{\circ}+\Delta a\ cos^2 6^{\circ}+y'\ sin\ 6^{\circ}+\Delta a\ sin^2 6^{\circ})^2}{a^2}+\frac{(-x'\ sin\ 6^{\circ}-\Delta a\ cos\ 6^{\circ}sin\ 6^{\circ}+y'\ cos\ 6^{\circ}+\Delta a\ sin\ 6^{\circ}cos\ 6^{\circ})^2}{b^2}=1$

$\frac{(x'\ cos\ 6^{\circ}+y'\ sin\ 6^{\circ}+\Delta a (cos^2 6^{\circ}+ sin^2 6^{\circ}) )^2}{a^2}+\frac{(-x'\ sin\ 6^{\circ}+y'\ cos\ 6^{\circ}+\Delta a (cos\ 6^{\circ}sin\ 6^{\circ}-cos\ 6^{\circ}sin\ 6^{\circ}))^2}{b^2}=1$

$\frac{(x'\ cos\ 6^{\circ}+y'\ sin\ 6^{\circ}+\Delta a )^2}{a^2}+\frac{(-x'\ sin\ 6^{\circ}+y''\ cos\ 6 ^{\circ})^2}{b^2}=1$

This shows the preservation of the vector length of the offset from the origin on the x axis in the frame of the DC section. The equation shows the ellipse rotated so that the vertex and co-vertex are no longer parallel to the frame of reference x and y axis.