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Circle Caustic

Consider a point light source located at a point $(\mu, 0)$. The Catacaustic of a unit Circle for the light at $\mu=\infty$ is the Nephroid

$\displaystyle x$ $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}[3\cos t-\cos(3t)]$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}[3\sin t-\sin(3t)].$ (2)

The Catacaustic for the light at a finite distance $\mu>1$ is the curve
$\displaystyle x$ $\textstyle =$ $\displaystyle {\mu(1-3\mu\cos t+2\mu\cos^3 t)\over -(1+2\mu^2)+3\mu\cos t}$ (3)
$\displaystyle y$ $\textstyle =$ $\displaystyle {2\mu^2\sin^3 t\over 1+2\mu^2-3\mu \cos t},$ (4)

and for the light on the Circumference of the Circle $\mu=1$ is the Cardioid
$\displaystyle x$ $\textstyle =$ $\displaystyle {\textstyle{2\over 3}} \cos t(1+\cos t)-{\textstyle{1\over 3}}$ (5)
$\displaystyle y$ $\textstyle =$ $\displaystyle {\textstyle{2\over 3}} \sin t(1+\cos t).$ (6)

If the point is inside the circle, the catacaustic is a discontinuous two-part curve. These four cases are illustrated below.

\begin{figure}\begin{center}\BoxedEPSF{CircleCaustic1.epsf scaled 500}\hskip 0.2in \BoxedEPSF{CircleCaustic2.epsf scaled 500}\end{center}\end{figure}

\begin{figure}\begin{center}\BoxedEPSF{CircleCaustic3.epsf scaled 500}\hskip 0.2in \BoxedEPSF{CircleCaustic4.epsf scaled 500}\end{center}\end{figure}


The Catacaustic for Parallel rays crossing a Circle is a Cardioid.

See also Catacaustic, Caustic




© 1996-9 Eric W. Weisstein
1999-05-26