Algorithmic solutions for the motion planning problem have been investigated\nfor five decades. Since the development of A* in 1969 many approaches have been\ninvestigated, traditionally classified as either grid decomposition, potential\nfields or sampling-based. In this work, we focus on using numerical\noptimization, which is understudied for solving motion planning problems. This\nlack of interest in the favor of sampling-based methods is largely due to the\nnon-convexity introduced by narrow passages. We address this shortcoming by\ngrounding the solution in differential geometry. We demonstrate through a\nseries of experiments on 3 Dofs and 6 Dofs narrow passage problems, how\nmodeling explicitly the underlying Riemannian manifold leads to an efficient\ninterior-point non-linear programming solution.\n