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.


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