See how pantoscopic tilt or face-form wrap changes a spectacle lens's power: the extra sphere and the cylinder it induces (Martin's formula), the compensated lens to order, and how far to lower the optical centre. For opticians and optometrists.
D
°
Often 8 to 12° on a well-fitted frame
Effective power when tilted
−10.10 −0.31 × 180
Nearest 0.25
−10.00 −0.25 × 180
To order (compensated, estimate)
−10.00 +0.25 × 180
Lower the optical centre by
5 mm
New sphere = −10.00 × (1 + sin²10° ÷ (2 × 1.498)) = −10.00 × 1.0101 = −10.10 D.
Induced cylinder = −10.00 × tan²10° = −10.00 × 0.0311 = −0.31 D, axis 180 (the lens turns about the horizontal).
Compensated: −10.00 ÷ 1.0101 = −9.90 D with +0.31 × 180 to cancel the induced cylinder; lens makers fine-tune wrap lenses by ray tracing.
1 mm lower for every 2° of pantoscopic tilt keeps the optical axis through the eye's centre of rotation.
For eye-care professionals. Martin's formula is for spherical lenses; with a prescribed cylinder it's a guide only.
Worked examples
−10.00 D at 10° pantoscopic tilt, n = 1.5. sin²10° = 0.0302, so the sphere becomes −10.00 × (1 + 0.0302 ÷ 3) = −10.10 D; tan²10° = 0.0311 gives a cylinder of −0.31 D. The lens acts as −10.10 −0.31 × 180, as in the NAO's worked example, and its optical centre should sit 5 mm lower.
−4.00 D polycarbonate in a 15° wrap frame. It acts as −4.08 −0.29 × 90. To get −4.00 D once wrapped, order about −4.00 +0.25 × 90 (to the nearest 0.25 D).
Formulas
New sphere: S′ = S × (1 + sin²θ ÷ 2n).
Induced cylinder: C = S × tan²θ, axis 180 for pantoscopic tilt and 90 for wrap. Some texts use S′ instead of S here, which gives a slightly larger cylinder; this calculator follows the NAO course.
Optical centre: lower it 1 mm for every 2° of pantoscopic tilt.
National Academy of Opticianry. Vertex Distance, Effective Power and Compensated Power / Tilt / Wrap (continuing-education course, 2025): Martin's formula and the −10.00 D at 10° example.
ABDO, OA Corner part 36: lowering the optical centre 1 mm per 2° of pantoscopic tilt.
Formulas checked on 19 October 2026. A guide for eye-care professionals: follow your lens maker's compensated values when given.
Frequently asked questions
What is Martin's formula?
A formula for the power of a spherical lens seen through obliquely, as happens when a lens is tilted in front of the eye: the sphere becomes S × (1 + sin²θ ÷ 2n) and a cylinder of S × tan²θ appears, where θ is the tilt and n the lens's refractive index. Both grow quickly with the angle and with the lens power.
What's the difference between pantoscopic tilt and wrap?
Pantoscopic tilt is the lens leaning forward at the bottom, which turns it about a horizontal line, so the induced cylinder has its axis at 180. Face-form wrap curves the frame round the face, which turns each lens about a vertical line, so the cylinder's axis is at 90.
Why lower the optical centre for pantoscopic tilt?
So that the lens's optical axis still passes through the centre of rotation of the eye. The usual rule is to lower the optical centre 1 mm for every 2° of pantoscopic tilt; this matters most for progressives, higher powers and aspheric lenses.
What does the compensated prescription mean?
The lens to order so that, once tilted, it gives the prescribed power: the sphere is divided by the same factor and a cylinder of the opposite sign cancels the induced one. It's a first-order estimate from Martin's formula; for wrap and freeform lenses, lens makers calculate the compensation by ray tracing, so use theirs when it's offered.
Does it work for a prescription with a cylinder?
Martin's formula is for spherical lenses. With a prescribed cylinder, or a lens tilted both ways at once, the effects combine in a more complicated way, so treat the result as a guide. Enter the sphere, or the power in the meridian that tilts, to see the size of the effect.
