CATACAUSTIC curves, in the higher geometry, that species of caustic curves which are formed by reflection.
These curves are generated after the following manner. If there be an infinite number of rays, as A B, A C, A 1), &c. (plate Miscellanies, fig. 6.) proceeding from the radiating point A, and reflect ed at any given curb B D H, so that the angles of incidence be still equal to those of reflection ; then the curve B E G, to which the reflected rays B I, C E, D F, &c. are tangents continually, as in the points I, E, F, is called the catacaus tic curve.
If the reflected I B be produced to K, so that A B B K, and the curve K L be the evolute of the catacaustic B E G, be ginning at the point K ; then the portion of the catacaustic BE= A C —A B X C E — B I continually. Or if any two in cident rays, as A B, A C be taken, that portion of the caustic that is evolved while the ray A B approaches to a coin cidence with A C, is equal to the differ ence of those incident rays X the differ ence of the reflected rays. When the given curve is a geometrical one, the catacaustic will be so too, and always rectifiable. The catacaustic of a circle is a cycloid, formed by the revolution of a circle along a circle. Thus, A B D,
fig. 7, being a semicircle exposed to parallel rays ; then those rays which fall near the axis C B will be reflected to F, the middle point of B C ; and those which fall at A, as they touch the curve only, will not be reflected at all ; but any inter mediate ray H I will be reflected to a point K, somewhere between A and F. And since every different incident ray will have a, different focal point, there fore, those various focal points will form a curve line A E F in one quadrant, and F G D 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 can dle light ; for then this curve A E F G D will appear plainly delineated on any white surface placed horizontally within the same, or on the surface of milk con tained in the bowl. The caustic of the common semi-cycloid, when the rays are parallel to the axis, is also a common cy cloid, described by the revolution of a circle upon the same base. The caustic of the logarithmic spiral is the same curve, only set in a different position.