Total curvature of a Caustic

Question

I am looking for a nudge in the right direction on the derivation of a formula for the Total Curvature of the Caustics to a manifold (a caustic is a planar family of curves reflected by a manifold).

Background - if the Fundamental Form is positive definite, the second fundamental form can be diagonalized with respect to it, and the eigenvalues are the principal curvatures $\kappa_1, \kappa_2$. One can then define the principal radii $R_1=\kappa^{-1}_{1}$ and $R_2 = \kappa^{-1}_{2}$. The product $\kappa_1 \kappa_2$ is the Total Curvature.

Now, I am asking for a reference to a formula I have been given anecdotally, but cannot find much about in the literature, that is, for the total curvature of the Caustic ($\mathcal{C}$):

$$K_{\mathcal{C}} = - (R_1 - R_2)^2\frac{\partial R_1 / \partial u}{\partial R_2 / \partial v} \qquad (*)$$

Where $(u, v)$ are the coordinates associated with the curvature lines. Does anyone have any experience with (*) or know of a derivation?

Answer 1

Q: "I am asking for a reference to a formula I have been given anecdotally".

A: This is equation (10) on page 340 in volume III of the classic text by Gaston Darboux, Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal (1887).
For an English text, see Differential Geometry: Manifolds, Curves, and Surfaces, page 398.
_{I do note that the formula in the OP is a bit different from what is given in these references, where the derivatives are taken with respect to the same variable in numerator and denominator.}