Caustic Hoodie

Shine light across a cup of coffee and a bright curved line appears on the surface, with a sharp point in the middle.

That line is a caustic: the envelope of the reflected rays, where neighbouring rays cross and pile up light.

For parallel light reflected inside a circle of radius R the caustic is half a nephroid, x = (R/4)(3cos t − cos 3t), y = (R/4)(3sin t − sin 3t). Its cusp sits at R/2 from the centre, the paraxial focus of a spherical mirror.

On the hoodie the cream and silver ring is the lit cup wall, the amber band is the caustic, and the bright point is the cusp.

We checked it by tracing 200,000 parallel rays: the crossing points of neighbouring reflected rays match the nephroid to better than one part in ten million of R.