  We show that two families of equations on the real line, the generalized
inviscid Proudman--Johnson equation, and the $r$-Hunter--Saxton equation
(recently introduced by Cotter et al.) coincide for a certain range of
parameters. This gives a new geometric interpretation of these
Proudman--Johnson equations as geodesic equations of right invariant
homogeneous $W^{1,r}$-Finsler metrics on an appropriate diffeomorphism group on
$\mathbb{R}$. Generalizing a construction of Lenells for the Hunter--Saxton
equation, we analyze the $r$-Hunter--Saxton equation using an isometry from the
diffeomorphism group to an appropriate subset of real-valued functions. Thereby
we show that the periodic case is equivalent to the geodesic equation on the
$L^r$-sphere in the space of functions, and the non-periodic case is equivalent
to a geodesic flow on a flat space. This allows us to give explicit solutions
to these equations in the non-periodic case, and answer several questions of
Cotter et al. regarding their limiting behavior.
