 Geodesic coordinate grid

For a surface metric $Ed{u}^2+2Fdudv+Gd{v}^2$, all coordinate curves are unparametrized geodesics precisely when the displayed identities hold. Parametrize $u=c$ by arc length, so $\dot{v}=G^{-1/2}$, and substitute into the transverse Euler-Lagrange equation for energy of a curve. This gives the first identity; $v=d$ gives the second. These identities concern geodesic images and do not assert that the coordinate variable itself is an affine parameter.