We propose a model of the functional architecture of curvature sensitive cells in the visual cortex that associates curvature with scale. The feature space of orientation and position is naturally enhanced via its oriented prolongation, yielding a 4-dimensional manifold endowed with a canonical Engel structure. This structure encodes position, orientation, signed curvature, and scale. We associate an open submanifold of the prolongation with the quasi-regular representation of the similitude group SIM(2), and find left-invariant generators for the Engel structure. Finally, we use the generators of the Engel structure to characterize curvature-sensitive receptive profiles.