Hold a well-filled cup of milky coffee on a sunny day and on the liquid’s surface you will see a catacaustic. This word has a Greek root and means ‘burning curve’. The sun is a point source at infinity whose parallel rays hit the cup according to the laws of geometrical optics. The rays are reflected from the reflective inner wall of the cup generating the bright curve, the caustic, formed by the envelope of the rays. The cusp at the centre of the caustic is called the paraxial focus and the liquid surface is brighter below the caustic curve. This particular shape of caustic is called nephroid, meaning kidney-shaped.
The phenomenon was known to Huygens in about 1679, and Bernoulli described it mathematically as an epicycloid. But almost two centuries earlier, Leonardo da Vinci observed the caustic when he was experimenting with using concave mirrors to generate heat — he called them fire mirrors. He argued that given equal diameters, the one with a shallower curve concentrates the most reflected rays, and ‘as a consequence, it will kindle a fire with greater rapidity and force’.
Syncline-shaped reflectors generate seismic reflections resembling these types of caustics.
Leonardo da Vinci’s early 16th-century sketch of a caustic, including his famous mirror writing.
This chapter is adapted from Carcione, J M (2007). Wave fields in Real Media, Elsevier. The cup image is original. The drawing is © The British Library Board, from Codex Arundel, MS 263, ff. 86v–87, ca. 1503–05, and used here with permission.