On the description of zonal aspheric surfaces

It is shown that, in general for a rotationally symmetric surface, the sagittal curvature is a function of the first derivative and tangential curvature is a function of the first and second derivatives of the surface equation. Smith and Atchison developed an "off-set" model for zonal aspheric surfaces with continuous gradients (or first derivatives) at the boundaries between zones of the surface, and discontinuous second derivatives at the boundaries. This model predicts continuous sagittal power errors but discontinuities in tangential power errors. However, measurements on commercially available zonal lenses by Atchison and Smith showed that both sagittal and tangential power errors were continuous across zone boundaries. Thus it is likely that the off-set model is incorrect. Zonal aspheric surfaces are more likely to be made according to a "co-axial" model which is described.