A **catacaustic **is a curve formed when light is reflected off of another curve, forming an envelope of reflected rays. If light is refracted instead, it is *diacaustic*.

For example, the nephroid is the catacaustic of the cardioid, if the light emanates from the cardioid’s cusp [1]:

Catacaustics were studied as test cases in the early development of calculus; Both Bernoulli and L’Hopital’s calculus works in the late 15th century included chapters on the topics.

The limaçon is the catacaustic of a circle when the light rays come from a point a finite (non-zero) distance from the circumference [2].

## Origins of catacaustic

The name catacaustic comes from *cata *(from Greek *catoptron*, mirror) and *caustic*:

Caustics, first studied by Tschirnhausen in 1682, are shimmering light patterns seen on the surface of reflective or refractive surfaces, like those formed on a lake in sunlight. They occur because sunlight reflects or refracts, converging at a point on a non-shiny surface [3]. Thus, catacaustics can be seen as a special case of caustics.

## References

[1] Baez, J. (2012). Rolling Circles and Balls (Part 2). Retrieved June 4, 2022 from:

[2] MacTutor. CC by SA 4.0. Retrieved June 6, 2022 from: https://mathshistory.st-andrews.ac.uk/Curves/Limacon/#:~:text=The%20limacon%20is%20an%20anallagmatic,de%20St%20Laurent%20in%201826.

[3] Garcia, E. (2005). ARTS 102 Aesthetics of the Algorithmic Image. Retrieved June 4, 2022 from: https://www.mat.ucsb.edu/~g.legrady/academic/courses/05f102/caustics.html