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?



Total curvature of a Caustic
**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](https://gallica.bnf.fr/ark:/12148/bpt6k778307/f348.item) (1887).  

For an English text, see [Differential Geometry: Manifolds, Curves, and Surfaces](https://archive.org/details/springer_10.1007-978-1-4612-1033-7/page/n409/mode/2up), 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.


