CATACAUSTICS

, or Catacaustic Curves, in the Higher Geometry, are the species of caustic curves formed by reflection.

These curves are generated after the following manner: If there be an infinite number of rays AB, AC, AD, &c, proceeding from the radiating point A, and reflected at any given curve BCDH, so that the angles of incidence be still equal to the angles of reflection; then the curve BEG, to which the reflect- ed rays BI, CE, DF, &c, are always tangents, as at the points I, E, F, &c, is the catacaustic, or causticby-reflection. Or it is the same thing as to say, that a caustic curve is that formed by joining the points of concurrence of the several reflected rays.

Some properties of these curves are as follow. If the reflected ray IB be produced to K, so that AB = BK, and the curve KL be the evolute of the caustic BEG, beginning at the point K; then the portion of the caustic BE is , that is, the difference of the two incident rays added to the difference of the two reflected rays.

When the given curve BCD is a geometrical one, the caustic will be so too, and will always be rectifiable. The caustic of the circle, is a cycloid, or epicycloid, formed by the revolution of a circle upon a circle.

Thus, ABD being a semicircle exposed to parallel rays; then those rays which fall near the axis CB will be reflected to F, the middle point of BC; and those which fall at A, as they touch the curve only, will not be reflected at all; but any intermediate ray HI will be reflected to a point K, somewhere between A and F. And since every different incident ray will have a different focal point, therefore those various focal points will form a curve line AEF in one quadrant, and FGD in the other, being the cycloid above-mentioned. And this figure may be beautifully exhibited experimentally by exposing the inside of a smooth bowl, or glass, to the sun beams, or strong candle light; for then this curve AEFGD will appear plainly delineated on any white surface placed horizontally within the same, or on the surface of milk contained in the bowl.

The caustic of the common cycloid, when the rays are parallel to its axis, is also a common cycloid, described by the revolution of a circle upon the same base. The caustic of the logarithmic spiral, is the same curve.

The principal writers on the caustics, are l'Hôpital, Carré, &c. See Memoires de l'Acad, an. 1666. & 1703. |