We call a surface parameterized by its principal curvatures if $\partial_1\nu = -\kappa_1\partial_1f$ and $\partial_2\nu=-\kappa_2\partial_2f$ holds for the normal $\nu$ and for principal curvatures $\kappa_1, \kappa_2$.
Is it always possible to reparameterize a surface such that the parameterization is by principal curvatures?
I would like to know whether this is true because I just proved the parallel surface area formula $$A(f^s) = A(f)-2s\int_UH(x,y)\sqrt{\det g(x,y)}d(x,y)+s^2\int_UK(x,y)\sqrt{\det g(x,y)}d(x,y)$$ where $H$ is the average and $K$ the Gaussian curvature if $f$ is parameterized by principal curvatures. I know this result holds in general, and I wonder whether the general result can be derived from this using reparameterization.