Geometry, Imaging and Computing
Volume 1 (2014)
$R$-transforms for Sobolev $H^2$-metrics on spaces of plane curves
Pages: 1 – 56
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.
plane curves, geodesic equation, Sobolev $H^2$-metric, $R$-transform
2010 Mathematics Subject Classification
35Q31, 58B20, 58D05