Vertex distance

Vertex distance

Vertex distanceis the distance between the back surface of a corrective lens, i.e.glasses (spectacles)orcontact lenses,and the front of thecornea.Increasing or decreasing the vertex distance changes the optical properties of the system, by moving thefocal pointforward or backward, effectively changing thepowerof the lens relative to theeye.Since mostrefractions(the measurement that determines the power of acorrective lens) are performed at a vertex distance of 12–14 mm, the power of the correction may need to be modified from the initial prescription so that light reaches the patient's eye with the same effective power that it did through thephoropterortrial frame.

Vertex distance is important when converting between contact lens and glasses prescriptions and becomes significant if the glasses prescription is beyond ±4.00diopters(often abbreviated D). The formula for vertex correction is$F_{c}=\left(F^{-1}-x\right)^{-1}$,where F_cis the power corrected for vertex distance, F is the original lens power, and x is the change in vertex distance in meters.

Derivation

The vertex distance formula calculates what power lens (F_c) is needed to focus light on the same location if the lens has been moved by a distancex.To focus light to the same image location:

$f_{c}=f-x$

wheref_cis the corrected focal length for the new lens,fis the focal length of the original lens, andxis the distance that the lens was moved. The value forxcan be positive or negative depending on the sign convention. Lens power in diopters is the mathematical inverse of focal length in meters.

${\begin{aligned}F&={\frac {1}{f}};&F_{\text{c}}&={\frac {1}{f_{\text{c}}}}\end{aligned}}$

Substituting for lens power arrives at

${\frac {1}{F_{\text{c}}}}={\frac {1}{F}}-x$

After simplifying the final equation is found:

${\begin{aligned}&&{\frac {F}{F_{\text{c}}}}&=1-xF\\&\Rightarrow &F_{\text{c}}&={\frac {F}{1-xF}}={\frac {1}{{\frac {1}{F}}-x}}\\&\Rightarrow &F&={\frac {1}{{\frac {1}{F_{\text{c}}}}+x}}\end{aligned}}$

Examples

Example 1: example prescription adjustment from glasses to contacts

A phoropter measurement of a patient reads −8.00D sphere and −5.25Dcylinderwith anaxisof 85° for one eye (the notation for which is typically written as−8 −5.25×85). The phoropter measurement is made at a common vertex distance of 12mm from the eye. The equivalent prescription at the patient's cornea (say, for a contact lens) can be calculated as follows (this example assumes anegativecylinder sign convention):

Power 1 is the spherical value, and power 2 is thesteeperpower of the astigmatic axis:

${\begin{aligned}{\text{Corrected power}}_{1}&=F_{{\text{c}}1}={\frac {1}{{\frac {1}{F}}-x}}={\frac {1}{{\frac {1}{-8}}-0.012}}=-7.30{\text{ D}},\\[3pt]{\text{Uncorrected power}}_{2}&=F_{\text{sphere}}+F_{\text{cylinder}}=-8+-5.25=-13.25{\text{ D}},\\[3pt]{\text{Corrected power}}_{2}&=F_{{\text{c}}2}={\frac {1}{{\frac {1}{F}}-x}}=-{\frac {13.25}{1-0.012(-13.25)}}=-11.43{\text{ D}},{\text{ and}}\\[3pt]{\text{Corrected cylinder}}&=F_{{\text{c}}2}-F_{{\text{c}}1}=-11.43-(-7.30)=-4.13{\text{ D}}.\end{aligned}}$

The axis value does not change with vertex distance, so the equivalent prescription for a contact lens (vertex distance, 0mm) is −7.30D of sphere, −4.13D of cylinder with 85° of axis (−7.30 −4.13×85or about−7.25 −4.25×85).

Example 2: example prescription adjustment from contacts to glasses

A patient has −8D sphere contacts. What is the equivalent prescription for glasses?

$F={\frac {1}{{\frac {1}{F_{\text{c}}}}+x}}={\frac {1}{{\frac {1}{-8}}+0.012}}=-8.84{\text{ D}}$

Therefore −8D contacts correspond to −8.75D or −9D glasses.

Example 3: sample plots

Corrected and uncorrected spherical power for a vertex distance of 12mm.

Difference in spherical power at a vertex distance of 12mm versus 0mm.

The following plots show the difference in spherical power at a 0mm vertex distance (at the eye) and a 12mm vertex distance (standard eyeglasses distance). 0mm is used as the reference starting power and is one-to-one. The second plot shows the difference between the 0mm and 12mm vertex distance powers. Above around 4D of spherical power, the difference versus the corrected power becomes more than 0.25D and is clinically significant.