## Nephroid Caustics

When you look into a mug you get light patterns at the bottom. The name given to these sharp lines caused by interference are caustics. I've been doing a bit of work trying to generate them using geometry alone to recreate the structure.

You can model the light going into the mug as a series of parallel lines which bounce once in the inside of the mug before hitting the bottom. Here's a diagram showing 4 rays come in and end somewhere in the right hand side:

The question is, what shape does the ends of these lines create as you change the length of the rays (this is equivalent to changing the depth of the mug, which you could do by drinking some of the liquid)?

For this I turned to Geogebra to try it out. Below I've shown a couple of results (there are a few dozen hidden construction lines and circles) for changing the length of the rays:

This shape is called a Nephroid and it is usually formed by rotating a circle of radius r/2 around one with radius r and tracking the path of a point on its circumference like this:

The mug takes half of this shape.