The Nephoid – a caustic in a coffee cup

The Nephoid is the curve formed by the envelope of rays reflected on a circular surface from a set of initial parallel rays.  In other words, the curve which owns all the rays as tangents.

The geomtery of each ray path is relatively straight forward, involving properties of parallel lines and isocelels triangles.

geometry of ray tracing

It is interesting to note that the ray path will only form a closed path for angles of \theta which divide $360^{o}$.

For a hemisphere, we would have,

caustic in semi circle

For a set of parallel rays enclosed in a circle we get:

caustic in enclosed circle

The envelope of the lines becomes clearer as the number of parallel rays are increased.

caustic in enclosed circle more rays

Conic sections in the sand give shipwrecked philosophers good hope


Conic sections in the sand give shipwrecked Socratic philosopher, Aristippus, good hope.

I wonder if this ancient tale, in which the mathematics of planetary motion some 2000 years prior to it’s application in space travel, is used to signify the presence of intelligent life and therefore hope of salvation has some echo in our modern era.

What obscure and impenetrable modern theorems could be left on a 21st century beach to do the same job?