Let $S$ be a regular surface with no parabolic or umbilical points. Let $\mathbf{x}: U \longrightarrow V$ be a parametrization of $S$ such that all the coordinates curves are also curvature lines. The parametrized surfaces:
$\begin{align*} \mathbf{y}(u,v) & = \mathbf{x}(u,v) + \rho_1 N(u,v), \\ \mathbf{z}(u,v) & = \mathbf{x}(u,v) + \rho_2 N(u,v), \end{align*}$
where $\rho_1=\dfrac{1}{k_1}$ and $\rho_2=\dfrac{1}{k_2}$ are called Focal Surfaces.
If $(k_1)_u$ and $(k_2)_v$ never vanish, how do I show that $\mathbf{y}$ and $\mathbf{z}$ are regular parametrized surfaces?