Geometry, Imaging and Computing

Volume 1 (2014)

Number 1

$R$-transforms for Sobolev $H^2$-metrics on spaces of plane curves

Pages: 1 – 56

DOI: http://dx.doi.org/10.4310/GIC.2014.v1.n1.a1

Authors

Martin Bauer (Fakultät für Mathematik, Universität Wien, Austria)

Martins Bruveris (Institut de mathématiques, EPFL, Lausanne, Switzerland)

Peter W. Michor (Fakultät für Mathematik, Universität Wien, Austria)

Abstract

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.

Keywords

plane curves, geodesic equation, Sobolev $H^2$-metric, $R$-transform

2010 Mathematics Subject Classification

35Q31, 58B20, 58D05

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