# caustic

## main

last updated: 2004-12-04

Generally spoken, a caustic 1) is formed by an envelope of light rays from a point source, diverted by some optical instrument. The source can be at finite distance (like a flame) or at infinite distance (like the sun).

Two situations can be distinguished:

Caustic qualities have been studied by Tschirnhausen (1682), Jacques Bernoulli (1691), and La Hire (1703).

The catacaustic of the logarithmic spiral is the curve itself (with a light source in the asymptote of the spiral). For a cycloid this is half true: for a cycloid arch, rays perpendicular to the x-axis result in two cycloid arches.

Some other catacaustic curves are:

 curve source catacaustic cardioid cusp nephroid circle on circumference cardioid not on circumference circle catacaustic at infinity nephroid cissoid of Diocles focus cardioid cycloid rays perpendicular to line through cusps cycloid (half size) deltoid infinity astroid equiangular spiral pole equiangular spiral exponential curve rays perpendicular y-axis catenary parabola infinity, rays perpendicular to the axis Tschirnhausen's cubic focus does not exist Pritch-Atzema spiral center circle quadrifolium center astroid Tschirnhausen's cubic pole semi-cubical parabola

The caustic can be generalized for different refraction indices (n1, n2) at the two sides (same and opposite side of S, in relation to the tangent). This curve is called the anticaustic.
For n (n1/n2) = 1 the orthotomic results.
It is said that the orthotomic can be identified with a cyclic.

Some anticaustic curves are the following:

 curve source anticaustic circle on circle Cartesian oval straight line symetric on line ellipse (n < 1) symetric on line hyperbola (n > 1)

The anticaustic curves have been studied by Mannheim.

notes

1) Kaustikos (Gr.) = what burns..