Generalization of the Minkwitz theorem to nonumbilical lines of symmetrical surfaces

The linear relation between the increase of peripheral astigmatism and the increase of power along an umbilical line was described by Minkwitz and is known as the Minkwitz theorem. However, in many cases, modern progressive addition lenses do not show an umbilical principal line. Therefore, we propose to extend the Minkwitz theorem to nonumbilical lines and higher order terms than the linear term of the increase in the peripheral astigmatism. We were able to derive a "generalized Minkwitz theorem," which holds true for a prescribed astigmatism at the principal line. The derived generalized Minkwitz theorem also indicates that the increase of the astigmatism perpendicular to the principal line depends not only on the power increase, as described by the Minkwitz theorem, but also on the astigmatism increase along the principal line. The Minkwitz theorem itself is a special case of this generalization.