 Unparametrized geodesic equation

A regular curve with tangent $v=dx/du$ is an unparametrized geodesic of an affine connection if ${\nabla }_{v}v=f(u)v$. Choose a new parameter $s$ with $ds/du=r(u)>0$ and $r^{\prime }/r=f$. Then $w=dx/ds=v/r$ obeys ${\nabla }_{w}w=r^{-2}({\nabla }_{v}v-(r^{\prime }/r)v)=0$. Conversely a reparametrized affine geodesic has acceleration proportional to its tangent. This criterion concerns the geometric curve, not a fixed parametrization.