Consider a straight river of width $L$ and two points $A$ and $B$ located on either side of it. How do you construct a bridge perpendicular to it to minimize the travel distance between $A$ and $B$?
Let $\vec{v}$ be the vector perpendicular to the river, directed towards $A$ and of length $L$. We denote $B^{\prime}$ as the image of the translation of $B$ according to $\vec{v}$. If the bridge connects $P$ and $Q$, then the path has a length of $A P + P Q + Q B = A P + L + P B^{\prime}$, so we want to minimize
$A P + P B^{\prime}$, which means placing $P$ at the intersection of $\left[A B^{\prime}\right]$ and the riverbank on the side of $A$.