Quotient elastic metrics on the manifold of arc-length parameterized plane curves

Tumpach Alice B.; Preston Stephen C.

doi:10.3934/jgm.2017010

Article Contents

2017, Volume 9, Issue 2: 227-256. Doi: 10.3934/jgm.2017010

This issue Previous Article The group of diffeomorphisms of the circle: Reproducing kernels and analogs of spherical functions Next Article

Quotient elastic metrics on the manifold of arc-length parameterized plane curves

Tumpach Alice B.^1, , and
Preston Stephen C.^2,

1.
Laboratoire Paul Painlevé, CNRS U.M.R. 8524, 59 655 Villeneuve d'Ascq Cedex, France
2.
Department of Mathematics, Brooklyn College and CUNY Graduate Center, New York, USA

* Corresponding author: Alice B. Tumpach
* Corresponding author: Alice B. Tumpach

Received: December 31, 2015

Revised: February 28, 2017

Published: October 2017

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Abstract

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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Mathematics Subject Classification: Primary: 53A04, 58B20; Secondary: 49Q20.

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Figure 1. Some parameterized closed immersions $\gamma$ in the plane

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Figure 2. Toy example: initial path joining a circle to the same circle via an ellipse. The 5 first shapes at the left correspond to the path at time $t = 0$, $t = 0.25$, $t = 0.5$, $t = 0.75$ and $t = 1$. The right picture shows the entire path, with color varying from red ($t=0$) to blue ($t = 0.5$) to red again ($t=1$)

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Figure 3. Straightening of the path illustrated in Fig. 2, with $a=100$ and $b=1$. The first line corresponds to the initial path, the second line to the path after 3500 iterations, and the third line corresponds to the path after 7000 iterations. Underneath, the evolution of the energy with respect to the number of iterations is depicted

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Figure 4. Negative gradient of the energy functional at the middle of the path depicted in Fig. 2 for $b=1$ and different values of the parameter $a/b$.

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Figure 5. Negative gradient of the energy functional at the middle of the path connecting a circle to the same circle via an ellipse for different values of the eccentricity of the middle ellipse. The first line corresponds to the values of parameters $a =0.01$ and $b=1$. The second line corresponds to $a = 100$ and $b=1$

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Figure 6. Negative gradient of the energy functional along the path depicted in Fig. 2 for $a=1$ (upper line), $a = 5$ (middle line) and $a = 50$ (lower line) and $b = 1$

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Figure 7. $2$-parameter family of variations of the middle shape of a path connecting a circle to the same circle

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Figure 8. Energy functional for the $2$-parameter family of paths whose middle shape is one of the shapes depicted in Fig. 7. The left upper picture corresponds to $a= 0.01$, $b =1$ and the right upper picture to $a = 100$, $b =1$. The lower picture shows the plots of both energy functionals with equal axis

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Figure 9. Different paths connecting a Mickey Mouse hand to the same hand with a missing finger

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Table 1. Energy of the paths depicted in Fig.9

parameters	lin. interpol. 1	lin. interpol. 2	path 3	path 4	path 5
$a=0.01$, $b = 1$	32.3749	27.45	25.3975	26.2504	28.3768
$a = 0.25$, $b =1$	63.1326	52.4110	47.8818	47.5037	48.2284
$a =100$, $b=1$	77.6407	66.6800	63.4840	60.9704	57.4557

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