  Let $S$ be a complete flat surface, such as the Euclidean plane. We determine
the homeomorphism class of the space of all curves on $S$ which start and end
at given points in given directions and whose curvatures are constrained to lie
in a given open interval, in terms of all parameters involved. Any connected
component of such a space is either contractible or homotopy equivalent to an
$n$-sphere, and every $n\geq 1$ is realizable. Explicit homotopy equivalences
between the components and the corresponding spheres are constructed.
