Caustics and Parametric Curves

Circle Catacaustic (Courtesy of Wolfram MathWorld)

If you have ever stared at a cylindrical glass of water sitting on table on a sunny day, you might have seen an intricate geometry of filaments of light projected onto the table behind the glass (in the direction away from the sun). This is called the caustic of the glass.

What exactly is a caustic? It is the envelope of rays from the reflection (catacaustic) or reflection (dicaustic) of light from an object. Actually the projected envelope from a glass is a nephroid (arising due to the differential equations involved) or a kidney-shaped object (see etymology).

It was this trivial experience, seeing something magical in a glass that started my obsession with optics/photonics. It also introduced me to the wonderful world of parametric curves, like the nephroid.

Parametric curves are interesting because they arise from very simple physical constraints (e.g. minimized degrees of freedom, path of least time, etc.). Consider the following problems:

Brachistochrone Problem
http://mathworld.wolfram.com/BrachistochroneProblem.html

Tautochrone Problem
http://mathworld.wolfram.com/TautochroneProblem.html

Other parametric curves such as involutes have important applications in engineering such as gears (so that a pair of gears mesh perfectly).
http://en.wikipedia.org/wiki/Involute_gear