  Non-relativistic particles that are effectively confined to two dimensions
can in general move on curved surfaces, allowing dynamical phenomena beyond
what can be described with scalar potentials or even vector gauge fields. Here
we consider a simple case of piecewise uniform curvature: a particle moves on a
plane with a spherical extrusion. Depending on the latitude at which the sphere
joins the plane, the extrusion can range from an infinitesimal bump to a nearly
full sphere that just touches the plane. Free classical motion on this surface
of piecewise uniform curvature follows geodesics that are independent of
velocity, while quantum mechanical scattering depends on energy. We compare
classical, semi-classical, and fully quantum problems, which are all exactly
solvable, and show how semi-classical analysis explains the complex quantum
differential cross section in terms of interference between two classical
trajectories: the sphere on a plane acts as a kind of double slit.
