Riemannian geometries on spaces of plane curves

Michor, Peter W., and Mumford, David. "Riemannian geometries on spaces of plane curves." Journal of the European Mathematical Society 008.1 (2006): 1-48. <http://eudml.org/doc/277745>.

@article{Michor2006,
abstract = {We study some Riemannian metrics on the space of smooth regular curves in the plane, viewed as the orbit space of maps from $S^1$ to the plane modulo the group of diffeomorphisms of $S^1$, acting as reparametrizations. In particular we investigate the metric, for a constant $A>0$, $G_c^A(h,k):=\int _\{S^1\}(1+A\kappa _c(\theta )^2)\langle h(\theta ),k(\theta )\rangle |c^\{\prime \}(\theta )|d\theta $ where $\kappa _c$ is the curvature of the curve $c$ and $h$, $k$ are normal vector fields to $c$. The term $A\kappa ^2$ is a sort of geometric Tikhonov regularization because, for $A=0$, the geodesic distance between any two distinct curves is 0, while for $A>0$ the distance is always positive. We give some lower bounds for the distance function, derive the geodesic equation and the sectional curvature, solve the geodesic equation with simple endpoints numerically, and pose some open questions. The space has an interesting split personality: among large smooth curves, all its sectional curvatures are $\ge 0$, while for curves with high curvature or perturbations of high frequency, the curvatures are $\le 0$.},
author = {Michor, Peter W., Mumford, David},
journal = {Journal of the European Mathematical Society},
keywords = {Riemannian metrics},
language = {eng},
number = {1},
pages = {1-48},
publisher = {European Mathematical Society Publishing House},
title = {Riemannian geometries on spaces of plane curves},
url = {http://eudml.org/doc/277745},
volume = {008},
year = {2006},
}

TY - JOUR
AU - Michor, Peter W.
AU - Mumford, David
TI - Riemannian geometries on spaces of plane curves
JO - Journal of the European Mathematical Society
PY - 2006
PB - European Mathematical Society Publishing House
VL - 008
IS - 1
SP - 1
EP - 48
AB - We study some Riemannian metrics on the space of smooth regular curves in the plane, viewed as the orbit space of maps from $S^1$ to the plane modulo the group of diffeomorphisms of $S^1$, acting as reparametrizations. In particular we investigate the metric, for a constant $A>0$, $G_c^A(h,k):=\int _{S^1}(1+A\kappa _c(\theta )^2)\langle h(\theta ),k(\theta )\rangle |c^{\prime }(\theta )|d\theta $ where $\kappa _c$ is the curvature of the curve $c$ and $h$, $k$ are normal vector fields to $c$. The term $A\kappa ^2$ is a sort of geometric Tikhonov regularization because, for $A=0$, the geodesic distance between any two distinct curves is 0, while for $A>0$ the distance is always positive. We give some lower bounds for the distance function, derive the geodesic equation and the sectional curvature, solve the geodesic equation with simple endpoints numerically, and pose some open questions. The space has an interesting split personality: among large smooth curves, all its sectional curvatures are $\ge 0$, while for curves with high curvature or perturbations of high frequency, the curvatures are $\le 0$.
LA - eng
KW - Riemannian metrics
UR - http://eudml.org/doc/277745
ER -