Calculating with optical correction power of parallel thin lens systems is very easy using the concept of vergence. However, if the lenses are not parallel, but tilted most of the calculations become complicated. A great lesson on the topic can be found here. A more scientifically rigorous derivation can be found in Harris W: Tilted Power of This lenses and Blendowske R. Oblique central refraction in tilted spherocylindrical lenses. Optom Vis Sci 2002;79:68–73.

These articles state that when a thin lens with power F is tilted at angle \(\theta\) than one get the following change in the dioptric power matrix: \(F_c = \Phi F \Phi\), where \(\Phi\) is the general tilt matrix with tilt angle \( \phi \) and tilt axis \( \theta \):

\(\Phi = \sqrt{h(\phi)} \left(I + (\frac{1}{\cos \phi} -1 ) \left(\begin{array}{cc}\sin^2 \theta & -\sin \theta \cos \theta\\-\sin \theta \cos \theta & \cos^2 \theta\end{array}\right) \right)\)

The derivation of the formula uses a third order Taylor-series approximation of trigonometric functions, which is pretty accurate with angles close to zero.

The special case of the formula is when the tilt angle is vertical \( \theta = \pi/2\) which is the pure wrap angle case and when the tilt angle is horizontal \( \theta = 0\), which is the pantoscopic angle case. In these cases the dioptric power matrix is diagonal, and hence the correction factor can be written in a simpler and more well known form:

\(F = F \left( 1+\frac{\sin^2 \theta}{2n} \right)\),

\(C = F \tan^2 \theta\).

Where \( F\) is the new vertex and \( C\) is the new cylinder.

It is clear from the above formulations that the pantoscopic and wrap angles need to be measured with respect to the focal points of the lens as compared the gaze vector, which is very tricky and difficult to fulfill. In practice, however, these calculations remain hidden from the optician since the lens designers and manufacturers using other -more clear- parameters for communication.

The typical definition that lens manufacturer gives to opticians for a pantoscopic angle is to measure the angle between the lens plane and the vertical. Here the assumption is that the customer’s gaze if horizontal.

For wrap angle (or face form) the definition is half of the angle of the frame to a straight line (and not the angle of the lens at the focal point). In theory, a lens manufacturer can calculate the correct wrap angle at the lookout point in front of the pupils once all the frame parameters are also given.

In summary, both of these angles give a (vertical or horizontal) cylinder to the lens which is very moderate for angles below 10 degrees angle.

\( sin^2(10) = 0,03\) and hence the correction ratio is only around 1% in the power of the lens. Similarly, the induced cylinder is only around 3% in the power of the lens. When the wrap angle or pantoscopic tilt goes above 20 degrees, then the effect becomes stronger and the induced cylinder can grow to 1 diopter in power for high power lens (above 6D).

It also follows from the above that the measurement of the wrap angle and the pantoscopic angle is vital only for high power lenses and high degrees in general, however, it is believed that frames with high tilt and or wrap angle are usually avoided by the opticians.