Focal Surface

  The impression of the caustic that we got in the last blog needs to be qualified to some extent. First the curves of constant ℓ and x are projections onto the x,y-plane but they are more like spirals in three dimensions. A photograph of a caustic would result from the flow of the sets of these evolving spirals through the frame of the photographic film during the time of the exposure. Hamilton chose a more limited definition of the surface of constant action based on a ray and those in its immediate vicinity which he called a pencil. However, we can use the prior results to play with statistics a little. The value of ℓ=2 did not give the the most compact image as it did for the focal point on the axis of the mirror but was close to it.

For both of the plots above the x,y-grid varied from (x,y)=(-1,-1) to (1,1) with steps of 0.1 in x and 0.2 in y.  As one increases the value of ℓ the focal point becomes more spread out, more blurred.

When one uses a smaller section of the mirror one gets a more well defined surface of constant action. The rays reflecting off the mirror centered at x=y=0 and with steps of 0.0001 in either direction produce a surface grid which gives following projection onto the x,y-plane. The red marker indicates the point that reflected off the mirror at x=y=-0.001. This appears to be what Hamilton referred to as a rectangular system. The horizontal lines are lines of constant x on the mirror. The vertical rows of markers indicate the lines of constant y on the mirror. The data is plotted relative to the mean values of x and y and ℓ was chosen to minimize the rms error.

So to study the system of rays and the focal properties of a mirror we need to think in terms of a flow field for the propagation of light.

edit (Jul 10): redid the last plot which also required some cleanup of the text.