Catacaustic
A catacaustic is a curve that is the envelope of rays emanating from a specified point (or a point at infinite distance producing parallel rays) for a given mirror shape. The point from which the rays emanate is called the radiant point. The catacaustic is an evolute of the orthotomic (Lawrence 1972, p. 60).
The following table lists the catacaustics for some common curves, omitting the incorrect catacaustic listed for the quadrifolium (Lawrence 1972, p. 207).
| curve | radiant point | catacaustic |
| cardioid catacaustic | cardioid cusp | nephroid |
| circle catacaustic | circumference | cardioid |
| circle catacaustic | parallel rays | nephroid |
| cissoid of Diocles catacaustic | focus | cardioid |
| cycloid catacaustic for 1 arch | parallel rays perpendicular to axis | 2 arches of a cycloid |
| deltoid catacaustic | parallel rays | astroid |
| ellipse catacaustic | any point | unnamed curve |
| natural logarithm catacaustic | parallel rays parallel to axis | catenary |
| logarithmic spiral catacaustic | origin | equal logarithmic spiral |
| parabola catacaustic | parallel rays perpendicular to axis | Tschirnhausen cubic |
| Tschirnhausen cubic catacaustic | focus | semicubical parabola |
Contact the MathWorld Team
12/17
