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Ellipse Caustic Curve

For an Ellipse given by

$\displaystyle x$ $\textstyle =$ $\displaystyle r\cos t$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle \sin t$ (2)

with light source at $(x, 0)$, the Caustic is
$\displaystyle x$ $\textstyle =$ $\displaystyle {N_x\over D_x}$ (3)
$\displaystyle y$ $\textstyle =$ $\displaystyle {N_y\over D_y},$ (4)

$\displaystyle N_x$ $\textstyle =$ $\displaystyle 2rx(3-5r^2)+(-6r^2+6r^4-3x^2+9r^2x^2)\cos t$  
  $\textstyle \phantom{=}$ $\displaystyle +6rx(1-r^2)\cos(2t)$  
  $\textstyle \phantom{=}$ $\displaystyle +(-2r^2+2r^4-x^2-r^2x^2)\cos(3t)$ (5)
$\displaystyle D_x$ $\textstyle =$ $\displaystyle 2r(1+2r^2+4x^2)+3x(1-5r^2)\cos t$  
  $\textstyle \phantom{=}$ $\displaystyle +(6r+6r^3)\cos(2t)+x(1-r^2)\cos(3t)$ (6)
$\displaystyle N_y$ $\textstyle =$ $\displaystyle 8r(-1+r^2-x^2)\sin^3 t$ (7)
$\displaystyle D_y$ $\textstyle =$ $\displaystyle 2r(-1-r^2-4x^2)+3(-x+5r^2)\cos t$  
  $\textstyle \phantom{=}$ $\displaystyle +6r(1-r^2)\cos(2t)+x(-1+r^2)\cos(3t).$ (8)

At $(\infty, 0)$,
$\displaystyle x$ $\textstyle =$ $\displaystyle {\cos t[-1+5r^2-\cos(2t)(1+r^2)]\over 4r}$ (9)
$\displaystyle y$ $\textstyle =$ $\displaystyle \sin^3 t.$ (10)

\begin{figure}\begin{center}\BoxedEPSF{EllipseCaustic1.epsf scaled 700}\hskip 0....
...\hskip 0.2 in\BoxedEPSF{EllipseCaustic3.epsf scaled 700}\end{center}\end{figure}

© 1996-9 Eric W. Weisstein