Clinical Consequences of Minkwitz’s Theorem

Take home message

Starting from Minkwitz’s theorem, many fundamental characteristics of progressive lenses can be deduced. Understanding the relationship between addition variation and astigmatism gives us a sense of lens optical geometry. In clinical practice, these considerations help to identify and understand some progressive lens fitting problems and to find a solution. ProCrea Tech has developed numerous occupational and high-end progressive lens designs that frequently solve these problems, meeting people’s visual needs.


Minkwiz’s theorem, first described in 1963, is a mathematical relationship that links addition variation with the resulting unintended astigmatism. Let’s explain it clearly. Imagining a map of a progressive lens, consider a small vertical segment on the medial line (or umbilical line) in the middle of the progressive corridor; let us also consider a horizontal segment of equal length starting from the medial line and going toward lens periphery. The change in cylinder power along the horizontal segment is equal to 2 times the change in addition along the vertical segment. In other words, the change in cylindrical power at a small distance away from the corridor is roughly equal to twice the change in addition power at an equal distance along the corridor.

However, the number ratio linking astigmatism and addition is not meaningful, as the theorem considered PAL geometries used at the time, which were simpler and less refined. Instead, the practical consequences of this mathematical relationship can be considered.

  • Firstly, although current PALs certainly have lower values of astigmatism growth, this ratio will never be 0. Thus, it is impossible to introduce an addition change in a surface without producing astigmatism.
  • The main consequence of Minkwitz’s theorem is that unwanted peripheral astigmatism is almost proportional to the addition power. Therefore, a lens with +2 addition, for example, will have about twice as much peripheral astigmatism as a lens with +1 addition.
  • Moreover, consequently to the theorem, a short progressive corridor corresponds to higher cylinder values (given the same addition).

Furthermore, the implications of the theorem are naturally detectable in clinical practice as well. In fact, usually lens areas that have an induced astigmatism lower than 0.50 D are considered the functional areas for vision. Therefore, as the addition increases (as the person ages), functional area width will decrease. If this is not taken into consideration a person might become a dissatisfied customer when upgrading their progressive eyeglass. It is important to consider these variations and for example, recommend occupational progressive lenses in some cases.

Similarly, functional areas will be reduced by choosing a smaller frame, which needs a short progressive corridor with more crowded aberrations. This factor may also be the cause of a failed PAL fitting, as the lens may not meet intermediate vision needs.

ProCrea Tech has developed numerous technologies and lens designs to address these issues, such as the high-end ANIMA design and a wide range of occupational PALs.

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