An Elementary Treatise on Optics/Chapter 4
CHAP. IV.
REFLEXION AT CURVED SURFACES NOT SPHERICAL.
17.It will be recollected that the common parabola has the peculiar property that two lines drawn from any point on the curve, one to the focus, the other parallel to the axis, make equal angles with the tangent or normal; whence it follows that rays proceeding from the focus of a paraboloid will be reflected accurately parallel to its axis, and vice versâ rays coming parallel to the axis will be made to converge accurately to the focus or from it according as it is the concave or convex surface that reflects.
18.In like manner rays proceeding from one focus of an ellipsoid will be reflected accurately to the other focus, or if the outward be the reflecting surface, rays converging towards one focus, will diverge after reflection as if they proceeded from the other.
19.The analogous property of the hyperbola leads one to the conclusion that the surface generated by the revolution of that figure about its major axis, is such that rays meeting in one focus, will after reflexion diverge from or converge to the other.
20. These are the only surfaces that reflect light accurately in the manner we have described,[1] which the simplest of all curved surfaces, the sphere, does not do; as to other surfaces, if we had occasion to treat of their reflexion, we should proceed in a manner similar to that used for the sphere, but in most instances if it were required to investigate the simple case of a small pencil of rays incident perpendicularly, it would be easiest to substitute for the surface its osculating sphere, that is, provided there were one, which is always the case at the vertex of a surface of revolution.
All this will, however, be much better understood after going through the next Chapter, which places the subject in a more general point of view.
- ↑ The solid generated by the revolution of the catenary about its axis reflects, pretty accurately, parallel rays not far from its axis, as that curve, for a short distance from its vertex, is very nearly parabolic.