Cylinder Transposition Calculator
Cylinder Transposition Calculator
Plus cylinder to minus cylinder and back. Three steps, and the axis is the one that goes wrong: 170 + 90 is 260, which must become 80.
Plus cyl ⇄ minus cyl
Axis wrap−1.00 +2.00 × 170 — the axis case that goes wrong
The three steps, and the wrap the first source omits
- step 1, the sphere
- quoted: “Add the sphere and cylinder powers to determine the new sphere power”. With signs: −1.00 plus +2.00 is +1.00, and +2.50 plus −1.50 is +1.00
- step 2, the cylinder
- quoted: “Change the sign of the cylinder”. The magnitude never changes — a 2.00 D cylinder is 2.00 D of astigmatism in either notation
- step 3, the axis, and the trap
- the first source read for this page says only “Change the axis by 90 degrees” and gives a worked example with an axis of 30, where adding 90 happens to work. It states no wrap rule at all. A second source states it explicitly: “If the original axis is less than or equal to 90 degrees then add 90” and “When original axis is greater than 90 degrees then subtract 90”. That is the rule this page implements
- why 170 + 90 is not 260
- the axis notation is modulo 180, because a cylinder axis is a meridian and a meridian at 260° is the same meridian as 80°. 260 − 180 = 80. The same arithmetic written as one expression is (axis + 89) mod 180 + 1, which maps 1 to 91, 90 to 180, 91 to 1, 170 to 80 and 180 to 90, and never leaves the range
- the self-check
- transposition is its own inverse. Doing it twice must return the original sphere, cylinder and axis exactly, and the spherical equivalent must be identical in both forms. Both are asserted here for every integer axis from 1 to 180 and for forty-two combinations of sphere and cylinder, including the boundary axes 1, 89, 90, 91, 170 and 180
- what transposition does not change
- the lens. The two forms are the same optical prescription written two ways, with the same spherical equivalent, the same two principal meridian powers and the same astigmatism. Nothing about transposing a prescription alters what a patient sees
Worked example
−1.00 +2.00 × 170 — the axis case that goes wrong
New sphere = −1.00 + (+2.00) = +1.00 D
New cylinder = −2.00 D — same magnitude, opposite sign
New axis: 170 is greater than 90, so subtract 90. 170 − 90 = 80°
The transposed prescription is +1.00 −2.00 × 80
The error this prevents. Add 90 instead and the axis comes out as 260°, which is not an axis. A system that prints it has produced an unmakeable prescription; one that silently clamps it to 180 has moved the cylinder 80° from where it belongs, which on a 2.00 D cylinder leaves the patient with most of their astigmatism uncorrected and a new induced cylinder on top
The check. Transpose +1.00 −2.00 × 80 again: sphere +1.00 + (−2.00) = −1.00, cylinder +2.00, axis 80 + 90 = 170. Back to the original, exactly
The boundary, both sides. Axis 89 goes to 179 (add); axis 90 goes to 180 (add — 90 is in the "90 or less" branch); axis 91 goes to 1 (subtract); axis 180 goes to 90. Axis 1 goes to 91. There is no axis 0 and no axis 181
The invariant. The spherical equivalent is −1.00 + (2.00 ÷ 2) = 0.00 D before and +1.00 + (−2.00 ÷ 2) = 0.00 D after. If those two disagree, something in the chain is wrong
The axis wrap at and around the boundary
| Axis entered | Branch of the rule | Axis + 90 | Transposed axis | Transposed twice |
|---|---|---|---|---|
| 1 | 90 or less — add | 91 | 91 | 1 |
| 45 | 90 or less — add | 135 | 135 | 45 |
| 88 | 90 or less — add | 178 | 178 | 88 |
| 89 | 90 or less — add | 179 | 179 | 89 |
| 90 | 90 or less — add | 180 | 180 | 90 |
| 91 | more than 90 — subtract | 181 | 1 | 91 |
| 92 | more than 90 — subtract | 182 | 2 | 92 |
| 135 | more than 90 — subtract | 225 | 45 | 135 |
| 170 | more than 90 — subtract | 260 | 80 | 170 |
| 180 | more than 90 — subtract | 270 | 90 | 180 |
Both forms of five prescriptions
| As written | Transposed | Meridian powers | Spherical equivalent |
|---|---|---|---|
| −1.00 +2.00 × 170 | +1.00 −2.00 × 80 | +1.00 and −1.00 | 0.00 D |
| −3.00 +2.00 × 30 | −1.00 −2.00 × 120 | −1.00 and −3.00 | −2.00 D |
| +2.50 −1.50 × 20 | +1.00 +1.50 × 110 | +1.00 and +2.50 | +1.75 D |
| −5.00 −2.50 × 90 | −7.50 +2.50 × 180 | −5.00 and −7.50 | −6.25 D |
| −0.75 −0.50 × 180 | −1.25 +0.50 × 90 | −1.25 and −0.75 | −1.00 D |
Three steps, two of which nobody gets wrong
Plus-cylinder and minus-cylinder notation describe the same lens. A sphero-cylinder has two principal meridians at right angles; you can write it as the flatter meridian’s power with a plus cylinder adding the difference, or as the steeper meridian’s power with a minus cylinder taking it away. Retinoscopes and keratometers produce the plus form naturally; most spectacle prescriptions in the United Kingdom and the United States are written in the minus form; and dispensing, contact-lens fitting and surgical planning all require moving between them. The conversion is three steps of arithmetic with no measurement involved.
Two of the three are trivial. Add the cylinder to the sphere, with signs, to get the new sphere. Flip the cylinder’s sign and keep its magnitude. The third is where implementations fail, because the axis is a modulo quantity and most statements of the rule do not say so. One of the two sources read for this page gives the step as “change the axis by 90 degrees” and illustrates it with an axis of 30, where adding 90 gives 120 and nothing goes wrong. It states no wrap rule anywhere. The other states it explicitly: add 90 if the axis is 90 or less, subtract 90 if it is more.
Why the subtraction is not a special case: a cylinder axis names a meridian, and a meridian has no direction. The axis notation therefore runs 1 to 180 and is arithmetic modulo 180, so 260° and 80° are the same meridian. Adding 90 to 170 gives 260, which is correct as an angle and invalid as an axis, and taking 180 off brings it back to the axis that was meant all along. The boundary deserves care in both directions: axis 90 transposes to 180, not to 0, because the convention has no axis 0; and axis 180 transposes to 90. An implementation tested only on a low axis will pass and still be wrong for every prescription above 90°, which is half of them.
There are two free checks and both are worth doing. Transposition is its own inverse, so transposing twice must return the original sphere, cylinder and axis exactly — a sign error in either of the first two steps, or a wrap error in the third, breaks it. And the spherical equivalent is identical in both forms, because adding the cylinder to the sphere and then subtracting half the flipped cylinder is the same as adding half the cylinder to the original sphere. Both are printed above. What transposition cannot do is change anything the patient experiences: the two forms have the same meridian powers, the same astigmatism and the same optics, and a transposed prescription is not a different prescription.
Frequently asked questions
How do I transpose a spectacle prescription?
Three steps. Add the cylinder to the sphere for the new sphere; keep the cylinder’s magnitude and flip its sign; and move the axis 90° — adding 90 if the axis is 90 or less, subtracting 90 if it is more. So −1.00 +2.00 × 170 becomes +1.00 −2.00 × 80.
Why is 170 + 90 not 260 for the axis?
Because an axis names a meridian, and a meridian at 260° is the same meridian as one at 80°. The notation runs from 1 to 180 and is modulo 180, so 260 − 180 = 80. An implementation that adds 90 and stops produces a number that is not an axis; one that clamps it to 180 puts the cylinder 80° from where it belongs.
What does axis 90 transpose to?
180. The rule’s add branch is “90 or less”, so 90 + 90 = 180, and 180 is a valid axis — the horizontal meridian. There is no axis 0 in the convention. Going the other way, 180 is more than 90, so it transposes to 90.
Does transposing change the lens?
No. The two forms are one prescription written two ways, with the same two principal meridian powers, the same amount of astigmatism and the same spherical equivalent. Nothing a patient sees depends on which notation the prescription is written in.
How can I check a transposition is right?
Two ways, both free. Transpose the answer again: it must return the original sphere, cylinder and axis exactly, because transposition is its own inverse. And compute the spherical equivalent of both forms: sphere plus half the cylinder must give the identical number. A sign error or a wrap error fails at least one of the two.
Related calculators
References
- Laramy-K Optical. Transposing Prescriptions. laramyk.com dispensing education. Accessed 9 October 2026.
- Optical Academy. Transpose Rx: How To Transpose An Eyeglass Prescription. optical-academy.com. Accessed 9 October 2026.
- Open Exam Prep. Vertex Distance, Sagittal Depth and Spherical Equivalent, Certified Lens Register study guide. open-exam-prep.com. Accessed 9 October 2026.
Not medical advice. For healthcare professionals and education. Reference intervals vary by laboratory and assay — always use your own laboratory's. Never base a dose or a treatment decision on this page alone. Full disclaimer at calcengines.com/disclaimer/