Contact Lens Vertex Calculator

Background.

Glasses rated −8.00 D sit about 12 mm in front of the eye. A lens of the same power placed directly on the cornea would over-correct that eye: the equivalent power at the cornea is about −7.30 D. A prescription number describes a lens at a particular distance from the eye, and moving the lens along the line of sight changes the power needed to bring light to the same focus.

Both lenses have the same job: to make light from a distant object converge towards, or seem to come from, one fixed point set by the eye, its far point. For a short-sighted eye that point lies in front of the eye, and a spectacle lens 12 mm forward is 12 mm closer to it, so it needs a shorter focal length and more minus power than a lens on the cornea. For a long-sighted eye the point lies behind the eye, the spectacle lens is further from it, and the lens on the eye needs the extra power. So people with strong short-sight wear contact lenses weaker than their glasses, and people with strong long-sight, including those who once wore thick glasses after cataract surgery without an implanted lens, need contacts stronger than their glasses.

The difference grows roughly with the square of the power. At ±2.00 D and 12 mm it is about 0.05 D, too small to matter against the usual quarter-dioptre steps in which lenses are made. At ±4.00 D it is approaching one of those steps, which is why that power is the usual point at which practitioners start allowing for it, and at ±10.00 D it is well over a dioptre.

The arithmetic here handles one spherical power at a time and is meant for understanding and checking. It is not a contact lens prescription: lenses must be prescribed and fitted by an eye-care professional, who also deals with astigmatism, the lens's fit on the cornea, the tear layer beneath it and how the lens actually performs on the eye.

What is contact lens vertex calculator?

Vertex distance is the gap between the back surface of a spectacle lens and the front of the cornea. Effective power, or vertex conversion, is the power a lens would need at a different position to produce the same vergence of light at the eye. It relies on vergence, the curvature of a wavefront measured in dioptres, which is the reciprocal of the distance in metres to the point the light is converging towards or diverging from. Moving a thin lens a known distance along the light path changes the vergence that reaches the eye in a predictable way, and the conversion undoes that change.

How to use this calculator.

1. 01Enter the known power in dioptres with its sign: minus for a short-sighted correction such as −6.50, plus for a long-sighted one such as +8.00.
2. 02Enter the distance the lens moves, in metres. Use a positive value when moving towards the eye, so spectacle plane to cornea at 12 mm is 0.012, and a negative value for the reverse, from cornea out to spectacles.
3. 03If you know the actual vertex distance of the frames, which an optician measures, use it rather than an assumed 12 mm; at high powers each millimetre counts.
4. 04Read the converted power, and keep the full figure until you compare it with the steps lenses are made in, so you can see which way it falls.

The formula.

F₂ = F₁ ÷ (1 − d × F₁), with F₁ the original power in dioptres, d the signed distance moved in metres, and F₂ the equivalent power at the new plane.

The derivation is short. A lens of power F₁ sends parallel light towards a focal point 1 ÷ F₁ metres away. Move a distance d towards that point and only 1 ÷ F₁ − d remains, so the vergence at the new plane is 1 ÷ (1 ÷ F₁ − d). Multiplying top and bottom by F₁ gives the formula.

The denominator explains the asymmetry. For a minus lens moved towards the eye, d × F₁ is negative, the denominator is greater than 1 and the power shrinks in magnitude. For a plus lens it is less than 1 and the power grows. For small d × F₁ the change is close to d × F₁², which is why it scales with the square of the power.

The relation treats the lens as infinitely thin and the light as paraxial, close to the optical axis. It breaks down when 1 − d × F₁ reaches zero, which happens when the new plane sits exactly at the lens's focal point; at a 12 mm shift that would take a +83.33 D lens, far beyond any prescription, and the calculation refuses that case.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Keating, Geometric, Physical, and Visual Optics, 2nd ed. (PRINT).
2. [2]University of Iowa, EyeRounds, Optics Review.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.