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

$$x = {\textstyle{1\over 4}}[3\cos t-\cos(3t)]$$ (1)

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

The Catacaustic for the light at a finite distance $\mu>1$ is the curve

$$x = {\mu(1-3\mu\cos t+2\mu\cos^3 t)\over -(1+2\mu^2)+3\mu\cos t}$$ (3)

$$y = {2\mu^2\sin^3 t\over 1+2\mu^2-3\mu \cos t},$$ (4)

and for the light on the Circumference of the Circle $\mu=1$ is the Cardioid

$$x = {\textstyle{2\over 3}} \cos t(1+\cos t)-{\textstyle{1\over 3}}$$ (5)

$$y = {\textstyle{2\over 3}} \sin t(1+\cos t).$$ (6)

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

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

See also Catacaustic, Caustic