Caustic

Remarks

Example: $Q = \R$ and $L = \set{(p^2, p) : p \in \R} \subset T^*\R = \R^2$. The map $\pi|_L$ is $p \mapsto p^2$, singular only at $p = 0$, so $\Sigma_L = \set{(0, 0)}$ and $C_L = \set{0}$. Over $q > 0$ lie two points of $L$, and over $q < 0$ none, so the number of preimages jumps at the caustic.

Non-example: if $L$ is the graph of $dS$ for a smooth function $S$ on $Q$, then $\pi|_L$ is a diffeomorphism, $\Sigma_L$ is empty, and the caustic is empty.

In geometric optics, $Q$ is space and $L$ is the set of positions and directions of the rays of a beam. Then $C_L$ is the envelope of the family of rays, where the rays focus. If $L$ is given locally by a generating family $F(q, y)$, the caustic is the bifurcation set of the family $y \mapsto F(q, y)$ with $q$ as control.