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) MR0115124 Zbl 05.0341.01 Zbl 05.0340.03 |
[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 MR0405513 |
[a4] | T. Poston, I. Stewart, "Catastrophe theory and its applications" , Pitman (1978) pp. Chapt. 12 MR0501079 Zbl 0382.58006 |
[a5] | J.W. Bruce, P.J. Giblin, "Curves and singularities: a geometrical introduction to singularity theory" , Cambridge Univ. Press (1984) MR1541053 Zbl 0534.58008 |
Caustic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Caustic&oldid=24393