= Geodesic coordinate grid
{title2=$(F/\sqrt G)_v=(\sqrt G)_u,\quad(F/\sqrt E)_u=(\sqrt E)_v$}
For a surface metric $E\,du^2+2F\,du\,dv+G\,dv^2$, 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.
Back to article page
