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Ellipse

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For the points $ P=(x,y)$ on an ellipse the sum of the distances from two fixed points $ F_{\pm}$ (the foci) is a given positive constant:

$\displaystyle \vert\overrightarrow{PF_-}\vert + \vert\overrightarrow{PF_+}\vert = 2 a $

with $ 2a>\vert\overrightarrow{F_-F_+}\vert$ .

\includegraphics[width=12.4cm]{a_ellipse}

If $ F_{\pm}=(\pm f,0)$ , then we have for the coordinates

$\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\quad b^2 = a^2 - f^2\, , $

and

$\displaystyle r^2 = \frac{b^2}{1-(f/a)^2\cos^2(\varphi)} $

for the polar coordinates of the points $ P$ .

A parameterisation of the ellipse is given by

$\displaystyle x=a\cos t,\quad y=b\sin t $

with $ t\in [0,2\pi)$ .

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  automatically generated 3/28/2008