The proximity force approximation (PFA) relates the interaction between
closely spaced, smoothly curved objects to the force between parallel plates.
Precision experiments on Casimir forces necessitate, and spur research on,
corrections to the PFA. We use a derivative expansion for gently curved
surfaces to derive the leading curvature modifications to the PFA. Our methods
apply to any homogeneous and isotropic materials; here we present results for
Dirichlet and Neumann boundary conditions and for perfect conductors. A Pad\'e
extrapolation constrained by a multipole expansion at large distance and our
improved expansion at short distances, provides an accurate expression for the
sphere-plate Casimir force at all separations.Comment: 4 pages, 1 figur