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

Homotopy type of spaces of curves with constrained curvature on flat surfaces