Bicycle paths, elasticae and sub-Riemannian geometry

We relate the sub-Riemannian geometry on the group of rigid motions of the plane to `bicycling mathematics. We show that this geometrys geodesics correspond to bike paths whose front tracks are either non-inflectional Euler elasticae or straight lines, and that its infinite minimizing geodesics (or `metric lines) correspond to bike paths whose front tracks are either straight lines or `Eulers solitons (also known as Syntractrix or Convicts curves).