Set theoretic, algebraic, mathematical morphology operations can be viewed as geometric deformations described by curve evolutions governed by a partial differential equation. While typical curve evolution implementations rely on embedding the curve as the zero level set of an evolving surface, this approach cannot handle open curves and junctions, and is inefficient due to the additional dimension. This paper presents a novel approach relying on an analytic wave propagation framework, but one which is implemented on a discrete grid and using discrete directions, by relying on subpixel geometric models. The resulting implementation is exact for the class of piecewise circular curves and its usefulness is demonstrated for extracting skeletons and for smoothing open curves and shapes.