Spherical Mirror Surface of Constant Path Length

  As mentioned previously for a spherical mirror the positions of a source image varies with changes in the relative position of the observer. As the rays to and from a reflection point vary with changes in the reflection point's position on the mirror the point on the final ray at constant total path length sweeps out a surface that is everywhere perpendicular to the rays.

In the plot above the three bent rays have a total path length equal to 5 units. The ends of the dashed lines indicate the positions of the apparent source images determined by the parallax from very small changes in position. The purple dashed line are normals to the final rays at Σℓ=5 units. Thus the surface of constant action will depend on the both the sum of path lengths for the rays and the shape of the surface of reflection. As the reflection point moves further away from the source the radius of curvature for the surface decreases.