Journal
JOURNAL OF GEOMETRIC MECHANICS
Volume 9, Issue 2, Pages 227-256Publisher
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/jgm.2017010
Keywords
Shape analysis of curves; quotient elastic metrics; minimisation of the energy functional
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Funding
- Labex CEMPI [ANR-11-LABX-0007-01]
- Simons Foundation [318969]
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We study the pull-back of the 2-parameter family of quotient elastic metrics introduced in [13] on the space of arc-length parameterized curves. This point of view has the advantage of concentrating on the manifold of arc-length parameterized curves, which is a very natural manifold when the analysis of un-parameterized curves is concerned, pushing aside the tricky quotient procedure detailed in [12] of the preshape space of parameterized curves by the reparameterization (semi-)group. In order to study the problem of finding geodesics between two given arc-length parameterized curves under these quotient elastic metrics, we give a precise computation of the gradient of the energy functional in the smooth case as well as a discretization of it, and implement a path-straightening method. This allows us to have a better understanding of how the landscape of the energy functional varies with respect to the parameters.
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