Calculus How To

Catacaustic: Definition, Examples

Share on

Calculus Handbook

Feel like "cheating" at Calculus? Check out our Practically Cheating Calculus Handbook, which gives you hundreds of easy-to-follow answers in a convenient e-book.

Calculus Curves >

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.


[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:,de%20St%20Laurent%20in%201826.
[3] Garcia, E. (2005). ARTS 102 Aesthetics of the Algorithmic Image. Retrieved June 4, 2022 from:

Stephanie Glen. "Catacaustic: Definition, Examples" From Calculus for the rest of us!

Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!

Leave a Reply

Your email address will not be published.