The Mirror Focal Points

  In Part I, section II.[10] of his Theory of Systems of Rays Hamilton connects the signs in the sum of the action with the shape of the mirrors and the nature of their focal points. An elliptical mirror has a real focus to which the rays converge and the sign of the length of the ray is taken as positive. For a hyperbolic mirror the rays diverge from the focus and the sign used is negative. We can use ℓ and ϑ to see what happens to the rays if these variables are changed slightly. The positions of the foci and the point of reflection allow us to determine the relations between the focal distances a and b and parameters ℓ and ϑ.

Independent adjustments to ℓ and ϑ allow us to find two nearby points on the elliptical and hyperbolic mirrors through the original point of reflection. Each pair of the new rays have new values of a and b which we can use to find a new reflection point (x,y). Placing the focal points on the x-axis simplifies the solution for the reflection point.

The values derived using the formulas starting with the parameters ℓ and ϑ check with the values obtained in the previous results. Plotting the rays for these reflections shows neighboring rays converge for the focus of the elliptical mirror and diverge for the hyperbolic mirror.

Supplemental (Jun 19): The parameters ℓ and ϑ form an auxiliary coordinate system derived from the biradial coordinates (a,b) for the focal points of the ellipse. For every pair of Cartesian coordinates (x,y) there is a unique pair of the auxiliary coordinates (ℓ, ϑ) in the plane of the rays. More generally the surfaces of reflection are ellipsoids and hyperboloids of revolution about the axis through the focal points. The auxiliary coordinates are related to prolate spheroidal coordinates. For both the ellipsoidal and hyperboloidal mirrors the angle of incidence equals the angle of reflection but while the reflected rays of the ellipsoidal mirror pass through the second focus those of the hyperboloid do not.