Consider a cusp catastrophe created by a screen as in the example and described by a standard cusp potential, Eq. (7.73). Suppose that a detector lies between the folds, so that there are three images of a single point source with state variables ã_i.(a) Explain how, in principle, it is possible to determine ã for a single image by measurements at D.

(b) Make a 3-dimensional plot of the location of the image(s) in ã - x̃ - ỹ space and explain why the names “fold” and “cusp” were chosen.(c) Prove as many as you can of the following scaling relations, valid in the limit as the cusp catastrophe is approached:

Eq. 7.73

3 ; = 0, = IN 3 =0, ;=0, i=1 i=l 3 3 *M=1, =0 and M= i=1 i=1 i=1 IN (7.82)