# Caustic

The envelope of rays reflected or refracted by a given curve. The caustic of reflected rays is called a catacaustic, that of refracted rays — a diacaustic.

Figure: c021000a

Figure: c021000b

For example, the catacaustic of a parallel beam of rays reflected from a semi-circle is part of an epicycloid (see Fig. a); the diacaustic of a pencil of rays emanating from a point lying in a denser medium and refracted by a straight line is part of an astroid with cusp the distance of which from the line is times that of the point (where is the refraction index) (see Fig. b).

#### References

[1] | A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian) |

#### Comments

In terms of purely geometric optics these are curves of light of infinite brightness consisting of points through which infinitely many reflected or refracted light rays pass. In reality they can often be observed as a pattern of pieces of very bright curves. E.g. on a sunny day at the seashore on the bottom beneath a bit of wavy water. Or at the bottom of a cup of tea into which light is shining. Suitably interpreted, caustics are bifurcation sets of elementary catastrophes with 3 control variables. To discuss not only the shape of caustics but also their brightness one needs wave optics and this leads to the study of asymptotic solutions of wave equations and oscillatory integrals [a2], [a3], [a4].

One way of describing the caustic resulting from rays from a point reflected by a curve is as follows. Take a point on and draw the tangent ; from draw the perpendicular to this tangent and let be the point on the perpendicular on the other side of at the same distance from as . Then the caustic by reflection is the evolute of the curve described by as runs over . This result is due to A. Quetelet. It follows readily that the caustic by reflection of the circle is the Pascal limaçon. There is a similar result for a caustic by refraction.

#### References

[a1] | G. Salmon, "Higher plane curves" , Hodges, Foster & Figgis (1879) |

[a2] | J. Chazarain, "Solutions asymptotiques et caustiques" F. Pham (ed.) , Rencontre de Congrès sur les singularités et leurs applications , Univ. Nice (1975) pp. 43–78 |

[a3] | J.J. Duistermaat, "Oscillatory integrals, Lagrange immersions, and unfolding of singularities" Comm. Pure Appl. Math. , 27 (1974) pp. 207–281 |

[a4] | T. Poston, I. Stewart, "Catastrophe theory and its applications" , Pitman (1978) pp. Chapt. 12 |

[a5] | J.W. Bruce, P.J. Giblin, "Curves and singularities: a geometrical introduction to singularity theory" , Cambridge Univ. Press (1984) |

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Caustic.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Caustic&oldid=11319