# Riemannian Geometries on Spaces of Plane Curves

@article{Michor2003RiemannianGO, title={Riemannian Geometries on Spaces of Plane Curves}, author={Peter W. Michor and David Mumford}, journal={Journal of the European Mathematical Society}, year={2003}, volume={8}, pages={1-48} }

We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of maps from the circle to the plane modulo the group of diffeomorphisms of the circle, acting as reparameterizations. In particular we investigate the L^2 inner product with respect to 1 plus curvature squared times arclength as the measure along a curve, applied to normal vector field to the curve. The curvature squared term acts as a sort of geometric Tikhonov regularization because…

## 395 Citations

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type Riemannian metrics on the space of planar curves

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Michor and Mumford have shown that the distances between planar curves in the simplest metric (not involving derivatives) are identically zero. We derive geodesic equations and a formula for…

Reparameterization Invariant Metric on the Space of Curves

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This paper uses the square root velocity function (SRVF) introduced by Srivastava et al. in [11] to define a reparameterization invariant metric on the space of immersions M' = Imm([0,1], M), and observes that such a natural choice of Riemannian metric on TM' induces a first-order Sobolev metric on M' with an extra term involving the origins, and leads to a distance.

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