Atchison D A, Smith G
Am J Optom Physiol Opt. 1986 Feb;63(2):156-62. doi: 10.1097/00006324-198602000-00012.
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.
结果表明，一般对于旋转对称表面，矢状曲率是一阶导数的函数，切向曲率是表面方程一阶和二阶导数的函数。史密斯和阿奇森针对在表面区域之间的边界处具有连续梯度（或一阶导数）且在边界处二阶导数不连续的区域非球面表面开发了一种“偏移”模型。该模型预测矢状屈光力误差是连续的，但切向屈光力误差存在不连续性。然而，阿奇森和史密斯对市售区域透镜的测量表明，矢状和切向屈光力误差在区域边界处都是连续的。因此，偏移模型很可能是不正确的。区域非球面表面更有可能是根据所描述的“同轴”模型制造的。
