Riemannian Geometries on Spaces of Plane Curves

By Peter W. Michor, David Mumford. 2003

443 citations recorded by the source corpus.

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, without it, the geodesic distance between any 2 distinct curves is 0, while in our case 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 positive or 0, while for curves with high curvature or perturbations of high frequency, the curvatures are negative.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2242da75b73536939d567cdf585e3c64ea17da20

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Citations in the corpus

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Global solutions to the volume-preserving mean-curvature flow (2015, background): “This gradient flow structure, however, is for the purpose of well-posedness results impracticle, since the L2 (geodesic) distance is degenerate in the sense that two well separated configurations may have zero L2 distance [24].”
Optimization Methods on Riemannian Manifolds and Their Application to Shape Space (2012, background): “(Note that simple L(2)-type metrics in the space of curves can in general not be used since the resulting spaces usually are degenerate [15]: They exhibit paths of arbitrarily small length between any two curves.”
Bootstrapping bilinear models of Simple Vehicles (2015, methodology): “The environment is a probability distribution on Maps that generates the map seen at each episode.”
Gromov–Wasserstein Distances and the Metric Approach to Object Matching (2011, background): “These developments have led to building Riemannian structures on many shape spaces [82–84, 115].”
INTRINSIC SHAPE ANALYSIS: GEODESIC PCA FOR RIEMANNIAN MANIFOLDS MODULO ISOMETRIC LIE GROUP ACTIONS (2007, methodology): “…methods have been proposed, e.g., by Miller, Trouvé, and Younes (2006) or Schmidt, Clausen and Cremers (2006), for general models, such as e.g., Michor and Mumford (2006) or Klassen et al. (2004), for closed 2D curves, or closed 2D and 3D contours based on medial axes, e.g., Pizer et al.…”
Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms (2004, background): “ions, so these spaces wrap up on themselves arbitrarily tightly, allowing the inﬁmum of path lengths between two given points to be zero. We also carry overto the generalcase the stronger metric from [10] which weights the L2 metric using the second fundamental form. It turns out that we have only t”
GEODESIC COMPLETENESS FOR SOBOLEV METRICS ON THE SPACE OF IMMERSED PLANE CURVES (2013, background): “Introduction Sobolev-type metrics on the space of plane immersed curves were independently introduced in [7, 17, 24].”