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

We consider the problem P curve of minimizing \(\int_{0}^{\ell} \sqrt{\beta^2 +|\kappa(s)|^2}{\rm d}s \) for a planar curve having fixed initial and final positions and directions. Here κ 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.