Cuspless sub-Riemannian geodesics within the Euclidean motion group SE(d)

R. Duits, A. Ghosh, T.C.J. Dela Haije, Y. Sachkov

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Abstract

We consider the problem P curve of minimizing TeX for a planar curve having fixed initial and final positions and directions. Here ¿ is the curvature of the curve with free total length l. This problem comes from a 2D model of geometry of vision due to Petitot, Citti and Sarti. Here we will provide a general theory on cuspless sub-Riemannian geodesics within a sub-Riemannian manifold in SE(d), with d¿=¿2, where we solve for their momentum in the general d-dimensional case. We will explicitly solve the curve optimization problem P curve in 2D (i.e. d¿=¿2) with a corresponding cuspless sub-Riemannian geodesic lifted problem defined on a sub-Riemannian manifold within SE(2). We also derive the solutions of P curve in 3D (i.e. d¿=¿3) with a corresponding cuspless sub-Riemannian geodesic problem defined on a sub-Riemannian manifold within SE(3). Besides exact formulas for cuspless sub-Riemannian geodesics, we derive their geometric properties, and we provide a full analysis of the range of admissible end-conditions. Furthermore, we apply this analysis to the modeling of association fields in neurophysiology
Original languageEnglish
Title of host publicationNeuromathematics of Vision
EditorsG. Citti, A. Sarti
Place of PublicationBerlin
PublisherSpringer
Pages173-215
ISBN (Print)978-3-642-34443-5
DOIs
Publication statusPublished - 2014

Publication series

NameLecture Notes in Morphogenesis
ISSN (Print)2195-1934

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Duits, R., Ghosh, A., Dela Haije, T. C. J., & Sachkov, Y. (2014). Cuspless sub-Riemannian geodesics within the Euclidean motion group SE(d). In G. Citti, & A. Sarti (Eds.), Neuromathematics of Vision (pp. 173-215). (Lecture Notes in Morphogenesis). Berlin: Springer. https://doi.org/10.1007/978-3-642-34444-2_5