Centre National de Recherche Scientifique (CNRS), CMAP, Ecole
Polytechnique, Route de Saclay, 91128
France, and Team GECO, INRIA-Centre de Recherche
2 Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands. Department of Mathematics and Computer Science
3 Aix-Marseille Univ, LSIS, 13013, Marseille, France
4 Program Systems Institute Pereslavl-Zalessky, Russia
Received: 31 July 2013
We consider the problem of minimizing 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.
Mathematics Subject Classification: 94A08 / 49J15
Key words: Curve reconstruction / generalized pontryagin maximum principle
The authors wish to thank Arpan Ghosh and Tom Dela Haije, Eindhoven University of Technology, for the contribution with numerical computations and figures.
This research has been supported by the European Research Council, ERC StG 2009 “GeCoMethods”, contract number 239748, by the ANR “GCM”, program “Blanc–CSD” project number NT09-504490, by the DIGITEO project “CONGEO”, by Russian Foundation for Basic Research, Project No. 12-01-00913-a, and by the Ministry of Education and Science of Russia within the federal program “Scientific and Scientific-Pedagogical Personnel of Innovative Russia”, contract No. 8209.
© EDP Sciences, SMAI, 2014