  Progressive addition lenses contain a surface of spatially-varying curvature,
which provides variable optical power for different viewing areas over the
lens. We derive complete compatibility equations that provide the exact
magnitude of cylinder along lines of curvatures on any arbitrary PAL smooth
surface. These equations reveal that, contrary to current knowledge, cylinder,
and its derivative, does not only depend on 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 only restricted to be applied
along lines of curvatures, but excluding umbilical points.
