Caustic
CAUSTIC , that which burns. In surgery, the term is given to substances used to destroy living tissues and so inhibit the action of organic poisons, as in bites, malignant disease and gangrenous processes. Such substances are silver nitrate (lunar caustic), the caustic alkalis (potassium and sodium hydrates), zinc chloride, an acid solution of mercuric nitrate, and pure carbolic acid. In mathematics, the "caustic surfaces" of a given surface are the envelopes of the normals to the surface, or the loci of its centres of principal curvature.
In optics, the term caustic is given to the envelope of luminous rays after reflection or refraction; in the first case the envelope is termed a catacaustic, in the second a diacaustic. Catacaustics are to be observed as bright curves when light is allowed to fall upon a polished riband of steel, such as a watch-spring, placed on a table, and by varying the form of the spring and moving the source of light, a variety of patterns may be obtained. The investigation of caustics, being based on the assumption of the rectilinear propagation of light, and the validity of the experimental laws of reflection and refraction, is essentially of a geometrical nature, and as such it attracted the attention of the mathematicians of the 17th and succeeding centuries, more notably John Bernoulli, G.F. de l'Hôpital, E.W. Tschirnhausen and Louis Carré.
Caustics by reflection
The simplest case of a caustic curve is when the reflecting surface is a circle, and the luminous rays emanate from a point on the circumference. If in fig. 1 AQP be the reflecting circle having C as centre, P the luminous point, and PQ any incident ray, and we join CQ it follows, by the law of the equality of the angles of incidence and reflection, that the reflected ray QR is such that the angles RQC and CQP are equal; to determine the caustic, it is necessary to determine the envelope of this line. This may be readily accomplished geometrically or analytically, and it will be found that the envelope is a cardioid (q.v.), i.e. an epicycloid in which the radii of the fixed and rolling circles are equal. When the rays are parallel, the reflecting surface remaining circular, the question can be similarly treated, and it is found that the caustic is an epicycloid in which the radius of the fixed circle is twice that of the rolling circle (fig. 2). The geometrical method is also applicable when it is required to determine the caustic after any number of reflections at a spherical surface of rays, which are either parallel or diverge from a point on the circumference. In both cases the curves are epicycloids; in the first case the radii of the rolling and the fixed circles are a(2n − 1)/4n and a/2n, and in the second, an/(2n + 1) and a/(2n + 1), where a is the radius of the mirror and n the number of reflections.
Fig. 1. Fig. 2. Fig. 3.
The Cartesian equation to the caustic produced by reflection at a circle of rays diverging from any point was obtained by Joseph Louis Lagrange; it may be expressed in the form
{ (4c − a) (x + y) ) − 2acx − ac } [3] = 27a [4]cy (x + y − c),
where a is the radius of the reflecting circle, and c the distance of the luminous point from the centre of the circle. The polar form is {(u + p) cos θ}2/3 + {(u − p) sin θ}2/3 = (2k)2/3, where p and k are the reciprocals of c and a, and u the reciprocal of the radius vector of any point on the caustic. When c = a or = ∞ the curve reduces to the cardioid or the two cusped epicycloid previously discussed. Other forms are shown in figs. 3, 4, 5, 6. These curves were traced by the Rev. Hammet Holditch (Quart. Jour. Math. vol. i.).
Fig. 4. Fig. 5.
Secondary caustics are orthotomic curves having the reflected or refracted rays as normals, and consequently the proper caustic curve, being the envelope of the normals, is their evolute. It is usually the case that the secondary caustic is easier to determine than the caustic, and hence, when determined, it affords a ready means for deducing the primary caustic. It may be shown by geometrical considerations that the secondary caustic is a curve similar to the first positive pedal of the reflecting curve, of twice the linear dimensions, with respect to the luminous point. For a circle, when the rays emanate from any point, the secondary caustic is a limaçon, and hence the primary caustic is the evolute of this curve.
Fig. 6.
Caustics by refraction
The simplest instance of a caustic by refraction (or diacaustic) is when luminous rays issuing from a point are refracted at a straight line. It may be shown geometrically that the secondary caustic, if the second medium be less refractive than the first, is an ellipse having the luminous point for a focus, and its centre at the foot of the perpendicular from the luminous point to the refracting line. The evolute of this ellipse is the caustic required. If the second medium be more highly refractive than the first, the secondary caustic is a hyperbola having the same focus and centre as before, and the caustic is the evolute of this curve. When the refracting curve is a circle and the rays emanate from any point, the locus of the secondary caustic is a Cartesian oval, and the evolute of this curve is the required diacaustic. These curves appear to have been first discussed by Gergonne. For the caustic by refraction of parallel rays at a circle reference should be made to the memoirs by Arthur Cayley.
References. - Arthur Cayley's "Memoirs on Caustics" in the Phil. Trans. for 1857, vol. 147, and 1867, vol. 157, are especially to be consulted. Reference may also be made to R.S. Heath's Geometrical Optics and R.A. Herman's Geometrical Optics (1900).
[1] Elie Bocthor (1784-1821) was a French orientalist of Coptic origin. He was the author of a Traité des conjugaisons written in Arabic, and left his Dictionary in MS.
Note - this article incorporates content from Encyclopaedia Britannica, Eleventh Edition, (1910-1911)
