  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).
