We introduce numerical schemes for computing distances and shortest paths with respect to several planar paths models, featuring curvature penalization and data-driven velocity: the Dubins car, the Euler/Mumford elastica, and a two variants of the Reeds-Shepp car. For that purpose, we design monotone and causal discretizations of the associated Hamilton-Jacobi-Bellman PDEs, posed on the three dimensional domain R 2 × S 1. Our discretizations involve sparse, adaptive and anisotropic stencils on a cartesian grid, built using techniques from lattice geometry. A convergence proof is provided, in the setting of discontinuous viscosity solutions. The discretized problems are solvable in a single pass using a variant of the Fast-Marching algorithm. Numerical experiments illustrate the applications of our schemes in motion planning and image segmentation.