The picture below is from Born & Wolf: Principles of Optics, 7th ed,, page 123, and is concerned with the propagation of optical wave fronts in a homogeneous medium.
The wave fronts are parallel in the sense that given one equi-phase surface (wave front), say, $\mathcal S_1$ then one can generate another by erecting the surface normal $\mathbf N(\mathcal P)$ on each point $\mathcal P \in \mathcal S_1$ and marking all points that are the same distance, say, $a$ from the foot $\mathcal P$ of the normal $\mathbf N(\mathcal P)$. These are the points $\mathcal Q \in \mathcal S_2$ where $\mathbf q = \mathbf p + a \mathbf N(\mathcal P)$ denote the position vectors corresponding to the points.
In the argument below, Born & Wolf are implying that if a point $\mathcal P$ is on either of the two lines of curvature then on any of its parallel surfaces, arbitrary $a$, the corresponding point $\mathcal Q$ will also lie on one of the lines of curvature, presumably the same min or max or curvature as the one on which the starting point is.
This sounds right but my question is how to prove this statement?
