Nonimaging Optics

Appendix K SKEW RAYS IN HYPERBOLOIDAL CONCENTRATOR

We shall prove that any ray aimed at the rim of the entry aperture is reflected to some point on this rim. Let FF′ in Figure K.1 be the diameter of the entry aperture, so that F and F′ are the foci of the hyperbolas in the plane of the diagram, and let P be a point on a hyperbola. The bisector PR of angle FPF′ is tangent to the hyperbola, and thus the tangent plane at P passes through PR. Any skew ray to the rim incident on the hyperbola at P passes through some point on the circle of which FF′ is the diameter and that is in a plane perpendicular to the diagram, and all such rays generate an oblique circular cone. We now draw F₁F₁′ through R to make the same angle with PR, as shown. A plane ...