## 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.

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,

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

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

## 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:

$1,2,3, \dots , 44$

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
$23 \rightarrow 2,\ 24 \rightarrow 4,\ 25 \rightarrow 8, \dots,\ 44 \rightarrow 44$

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