Progressive addition lenses (PALs) contain a surface of spatially varying curvature, which supplies variable optical power for different viewing areas over the lens. We derive complete compatibility equations providing the exact magnitude of a cylinder along lines of curvature on any arbitrary PAL smooth surface. These equations reveal that, contrary to current knowledge, the cylinder and its derivative depend not only on the principal curvature and its derivatives along the principal line but also on the geodesic curvature and its derivatives along the line orthogonal to the principal line. We quantify the relevance of the geodesic curvature through numerical computations. We also derive an extended and exact Minkwitz theorem restricted only to be applied along lines of curvature, but excluding umbilical points.