Interpolating smooth surfaces from boundary conditions is a ubiquitous problem in early visual processing. We describe a solution for an important special case: the interpolation of surfaces that are locally spherical or cylindrical from initial orientation values and constraints on orientation. The approach exploits an observation that components of the unit normal vary linearly on surfaces of uniform curvature, which permits implementation using local parallel processes. Experiments on spherical and cylindrical test cases have produced essentially exact reconstructions, even when boundary values were extremely sparse or only partially constrained. Results on other test cases seem in reasonable agreement with human perception.