The Nephoid – a caustic in a coffee cup

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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

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