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| 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
{ (4c² − a²) (x² + y²) ) − 2a²cx − a²c² }3 = 27a4c²y² (x² + y² − c²)²,
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 ½θ}2/3 + {(u − p) sin ½θ}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.).
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| 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. |
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 Caustics by refraction. 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).