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yungman
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This is really a simplification of an equation. It is just very long and complex and I need advice.
For ellipse center at the origin with tilt angle [itex]\tau[/itex], the distance from origin to the ellipse is [itex]\rho(\varsigma)[/itex].
It is given:
[tex]\varsigma\;=\;\frac {y}{x}\;=\;\frac{2E_xE_y\cos\delta}{E_x^2-E_y^2}\;,\; a=\frac{1}{E_x^2\sin^2\delta}\;,\;b=\frac{2\cos \delta}{E_x E_y \sin^2\delta}\;,\;c=\frac {1}{E^2_y\sin^2\delta}[/tex]
[tex]x^2=\frac{1}{c\varsigma^2-b\varsigma+a}[/tex]
The equation of ellipse is:
[tex] \vec E (0,t)\;=\;\hat x E_x\cos(\omega t)\;+\;\hat y E_y\cos(\omega t +\delta)[/tex]
Where [itex] E_x,\;E_y, \; \delta[/itex] are all constant.
[tex]x=E_x\cos(\omega t)\;\hbox{ and }\; y=E_y\cos(\omega t +\delta)[/tex]
[tex]\rho^2(\varsigma)\;=\; x^2+y^2\;=\;x^2(1+\varsigma^2)\;=\;\frac {(1+\varsigma^2)}{c\varsigma^2-b\varsigma+a}[/tex]The major axis [itex]\rho_{max}[/itex]=OA is given by the book where:
[tex]OA=\sqrt{\frac{1}{2}[E^2_x+E^2_y+\sqrt{E_x^4+E_y^4+2E_x^2E_y^2\cos(2 \delta)}}[/tex]
I tried to substitute everything in, it get way complicated and no where close. It is just too long to type in my attempt. Can anyone suggest a way to simplify this?
Thanks
For ellipse center at the origin with tilt angle [itex]\tau[/itex], the distance from origin to the ellipse is [itex]\rho(\varsigma)[/itex].
It is given:
[tex]\varsigma\;=\;\frac {y}{x}\;=\;\frac{2E_xE_y\cos\delta}{E_x^2-E_y^2}\;,\; a=\frac{1}{E_x^2\sin^2\delta}\;,\;b=\frac{2\cos \delta}{E_x E_y \sin^2\delta}\;,\;c=\frac {1}{E^2_y\sin^2\delta}[/tex]
[tex]x^2=\frac{1}{c\varsigma^2-b\varsigma+a}[/tex]
The equation of ellipse is:
[tex] \vec E (0,t)\;=\;\hat x E_x\cos(\omega t)\;+\;\hat y E_y\cos(\omega t +\delta)[/tex]
Where [itex] E_x,\;E_y, \; \delta[/itex] are all constant.
[tex]x=E_x\cos(\omega t)\;\hbox{ and }\; y=E_y\cos(\omega t +\delta)[/tex]
[tex]\rho^2(\varsigma)\;=\; x^2+y^2\;=\;x^2(1+\varsigma^2)\;=\;\frac {(1+\varsigma^2)}{c\varsigma^2-b\varsigma+a}[/tex]The major axis [itex]\rho_{max}[/itex]=OA is given by the book where:
[tex]OA=\sqrt{\frac{1}{2}[E^2_x+E^2_y+\sqrt{E_x^4+E_y^4+2E_x^2E_y^2\cos(2 \delta)}}[/tex]
I tried to substitute everything in, it get way complicated and no where close. It is just too long to type in my attempt. Can anyone suggest a way to simplify this?
Thanks
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