The Ellipse

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Viewing the conic section ϕ 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

x2a2+y2b2=1

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


(x+h)2a2+(y+k)2b2=1


In the plane of the DC sector, this equation becomes


(x+Δa)2a2+(y)2b2=1

where the center of the ellipse is found at {Δa,0}.


Switching to the frame of the wires, the ellipse is still centered at {Δa,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 θ, with the rotation matrix R(θyx). In the frame of the wires, this center point falls at


[xyz]=[cos 6sin 60sin 6cos 60001][xyz]



[xyz]=[cos 6sin 60sin 6cos 60001][Δa00]
[xyz]=[Δa cos 6Δa sin 60]
\(x",y,z)Center\)=(Δa cos 6,Δa sin 6,0)=\(h,k,0\)
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 [math]6^{\circ}[/math] counterclockwise from the x axis, with the intersection points having a uniform spacing in the ellipse parameter.


(x y z

)=(cos 6\[Degree] sin 6\[Degree] 0 -sin 6\[Degree] cos 6\[Degree] 0 0 0 1

) . (x' y' z'

)


(x y z

)=(x'cos 6\[Degree]+y' sin 6\[Degree] -x'sin 6\[Degree]+y'cos 6\[Degree] 0

)

Substituting this into the equation for an ellipse in the frame of the wires,

(x+h)^2/a^2+(y+k)^2/b^2=1


((x'+\[CapitalDelta]a Cos[6 \[Degree]])Cos[6 \[Degree]]+(y'+\[CapitalDelta]a Sin[6 \[Degree]])Sin[6 \[Degree]])^2/a^2+(-(x'+\[CapitalDelta]a Cos[6 \[Degree]])Sin[6 \[Degree]]+(y'+\[CapitalDelta]a Sin[6 \[Degree]])Cos[6 \[Degree]])^2/b^2=1

(x'Cos[6 \[Degree]]+\[CapitalDelta]a (Cos^2)[6 \[Degree]]+y'Sin[6 \[Degree]]+\[CapitalDelta]a (Sin^2)[6 \[Degree]])^2/a^2+(-x'Sin[6 \[Degree]]-\[CapitalDelta]a Cos[6 \[Degree]]Sin[6 \[Degree]]+y'Cos[6 \[Degree]]+\[CapitalDelta]a Sin[6 \[Degree]]Cos[6 \[Degree]])^2/b^2=1

(x'Cos[6 \[Degree]]+y'Sin[6 \[Degree]]+\[CapitalDelta]a (Cos^2)[6 \[Degree]]+\[CapitalDelta]a (Sin^2)[6 \[Degree]])^2/a^2+(-x'Sin[6 \[Degree]]+y'Cos[6 \[Degree]]+\[CapitalDelta]a Cos[6 \[Degree]]Sin[6 \[Degree]]-\[CapitalDelta]a Cos[6 \[Degree]]Sin[6 \[Degree]])^2/b^2=1

(x'Cos[6 \[Degree]]+y'Sin[6 \[Degree]]+\[CapitalDelta]a ((Cos^2)[6 \[Degree]]+ (Sin^2)[6 \[Degree]]))^2/a^2+(-x'Sin[6 \[Degree]]+y'Cos[6 \[Degree]]+\[CapitalDelta]a (Cos[6 \[Degree]]Sin[6 \[Degree]]-Cos[6 \[Degree]]Sin[6 \[Degree]]))^2/b^2=1

(x'Cos[6 \[Degree]]+y'Sin[6 \[Degree]]+\[CapitalDelta]a )^2/a^2+(-x'Sin[6 \[Degree]]+y'Cos[6 \[Degree]])^2/b^2=1