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

The Cardioid as the Envelope of Pencil Curves

Draw a circle and divide the circumference into and even number of equally spaced points.  In this example I have used 44:

1,2,3, \dots , 44

image 1 circle

Draw following chords between numbers in such a way that when viewed as a sequemce, the second point of each pair moves twice as fast around the circle.:
1 \rightarrow 2,\ 2 \rightarrow 4,\ 4 \rightarrow 8, \dots,\ 22 \rightarrow 44
and
23 \rightarrow 2,\ 24 \rightarrow 4,\ 25 \rightarrow 8, \dots,\ 44 \rightarrow 44

image 2 lines

and on completion, sketch the curve which owns each of the chords as a tangent:

image 3 completed