Geodesic
Curve cuspless reconstruction via sub-riemannian geometry
ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 3, pp. 748-770

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We consider the problem of minimizing ∫ 0 ℓ ξ 2 +K 2 (s)ds ∫ 0 ℓ ξ 2 + K 2 ( s )   d s for a planar curve having fixed initial and final positions and directions. The total length ℓ is free. Here s is the arclength parameter, K(s) is the curvature of the curve and ξ > 0 is a fixed constant. This problem comes from a model of geometry of vision due to Petitot, Citti and Sarti. We study existence of local and global minimizers for this problem. We prove that if for a certain choice of boundary conditions there is no global minimizer, then there is neither a local minimizer nor a geodesic. We finally give properties of the set of boundary conditions for which there exists a solution to the problem.

DOI : 10.1051/cocv/2013082
Classification : 94A08, 49J15
Keywords: curve reconstruction, generalized pontryagin maximum principle 
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     author = {Boscain, Ugo and Duits, Remco and Rossi, Francesco and Sachkov, Yuri},
     title = {Curve cuspless reconstruction \protect\emph{via }sub-riemannian geometry},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {748--770},
     publisher = {EDP-Sciences},
     volume = {20},
     number = {3},
     year = {2014},
     doi = {10.1051/cocv/2013082},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013082/}
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Boscain, Ugo; Duits, Remco; Rossi, Francesco; Sachkov, Yuri. Curve cuspless reconstruction via sub-riemannian geometry. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 3, pp. 748-770. doi: 10.1051/cocv/2013082

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