Parallax and the Focal Point of the Reflected Rays

  Hamilton's description of a surface of constant action is contained in III [13] and uses the sum of the separate pathlengths. The system of final reflected rays all focus on an image of the luminous source. This is why the method of images used in the last blog works. One can illustrate this with a simple problem. The first row of data below contains the position of the source, the nearest point on the mirror, the normal to the mirror and the reflected image of the source. The second row contains two points, f̄ and ḡ, used to select the rays, their reflected images and the points of reflection on the mirror. The next row gives the bent rays and the final row gives the ray extensions to the source image.

The plot shows the shift in the direction of the apparent source due to parallax associated with changes in the position of observation.

The green line represents the arc whose action or path length is the same as that of f̄ and the direction of the fnal rays to the source image are all perpendicular to this surface. For reflection the surfaces of constant action are spheres due to the spherical propagation of light.