  We consider spaces of smooth immersed plane curves (modulo translations
and/or rotations), equipped with reparameterization invariant weak Riemannian
metrics involving second derivatives. This includes the full $H^2$-metric
without zero order terms. We find isometries (called $R$-transforms) from some
of these spaces into function spaces with simpler weak Riemannian metrics, and
we use this to give explicit formulas for geodesics, geodesic distances, and
sectional curvatures. We also show how to utilise the isometries to compute
geodesics numerically.
