2013/05/26 by Remco Duits, Duits, Remco, Arpan Ghosh +5
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Advanced Differential Geometry Research #Analytic and geometric function theory
paper · pdf · doi:10.48550/arxiv.1305.6061
We consider the problem mathbfPcurve of minimizing \∫ \0L\n\√(\ξ2 + \κ2(s)) , rm ds for a curve \x on mathbb R\nwith fixed boundary points and directions. Here the total length L\≥ 0 is\nfree, s denotes the arclength parameter, \κ denotes the absolute\ncurvature of \x, and \ξ>0 is constant. We lift problem\n mathbfPcurve on mathbb R3 to a sub-Riemannian problem\n mathbfPmec on \SE(3) nolimits/( \0 \×\n\SO(2) nolimits). Here, for admissible boundary conditions, the\nspatial projections of sub-Riemannian geodesics do not exhibit cusps and they\nsolve problem mathbfPcurve. We apply the Pontryagin Maximum Principle\n(PMP) and prove Liouville integrability of the Hamiltonian system. We derive\nexplicit analytic formulas for such sub-Riemannian geodesics, relying on the\nco-adjoint orbit structure, an underlying Cartan connection, and the matrix\nrepresentation of \SE(3) nolimits arising in the Cartan-matrix.\nThese formulas allow us to extract geometrical properties of the sub-Riemannian\ngeodesics with cuspless projection, such as planarity conditions, explicit\nbounds on their torsion, and their symmetries. Furthermore, they allow us to\nparameterize all admissible boundary conditions reachable by geodesics with\ncuspless spatial projection. Such projections lay in the upper half space. We\nprove this for most cases, and the rest is checked numerically. Finally, we\nemploy the formulas to numerically solve the boundary value problem, and\nvisualize the set of admissible boundary conditions.\n