In the first part, we revisit some key notions. Let M be a Riemannian manifold. Let G be a group acting on M. We discuss the relationship between the quotient M∕G, “horizontality” and “normalization”. We discuss the distinction between path-wise invariance and point-wise invariance and how the former positively impacts the design of metrics, in particular for the mathematical and numerical treatment of geodesics. We then discuss a strategy to design metrics with desired properties.
In the second part, we prepare methods to normalize some standard group actions on the curve; we design a simple differential operator, called the delta operator, and compare it to the usual differential operators used in defining Riemannian metrics for curves.
In the third part we design two examples of Riemannian metrics in the space of planar curves. These metrics are based on the “delta” operator; they are “modular”, they are composed of different terms, each associated to a group action. These are “strong” metrics, that is, smooth metrics on the space of curves, that is defined as a differentiable manifolds, modeled on the standard Sobolev space H2. These metrics enjoy many important properties, including: metric completeness, geodesic completeness, existence of minimal length geodesics. These metrics properly project on the space of curves up to parameterization; the quotient space again enjoys the above properties.
Accepté le :
DOI : 10.1051/cocv/2018044
Mots-clés : Riemannian metric, manifold of curves, shape space, Hilbert manifold, Sobolev space, geodesic, Fréchet mean
@article{COCV_2019__25__A59_0,
author = {Mennucci, Andrea C.G.},
title = {Designing metrics; the delta metric for curves},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2018044},
mrnumber = {4023127},
zbl = {1437.49059},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2018044/}
}
TY - JOUR AU - Mennucci, Andrea C.G. TI - Designing metrics; the delta metric for curves JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2018044/ DO - 10.1051/cocv/2018044 LA - en ID - COCV_2019__25__A59_0 ER -
Mennucci, Andrea C.G. Designing metrics; the delta metric for curves. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 59. doi : 10.1051/cocv/2018044. http://archive.numdam.org/articles/10.1051/cocv/2018044/
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