Ellipse Caustic Curve

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

, the Caustic is

$\displaystyle x$	$\textstyle =$	$\displaystyle {N_x\over D_x}$	(3)
$\displaystyle y$	$\textstyle =$	$\displaystyle {N_y\over D_y},$	(4)

where

$\displaystyle N_x$	$\textstyle =$	$\displaystyle 2rx(3-5r^2)+(-6r^2+6r^4-3x^2+9r^2x^2)\cos t$
	$\textstyle \phantom{=}$	$\displaystyle +6rx(1-r^2)\cos(2t)$
	$\textstyle \phantom{=}$	$\displaystyle +(-2r^2+2r^4-x^2-r^2x^2)\cos(3t)$	(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
1999-05-25