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 geometry's 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 'Euler's solitons' (also known as syntractrix or convicts' curves).