A prescription with astigmatism can be written two ways: with a plus cylinder or a minus cylinder. Some prescribers use one form and some the other, while labs and lensmeter work commonly use minus-cylinder form. Being able to switch between them quickly — and know they mean the same lens — is a core skill for opticians and ophthalmic assistants alike.
The three steps
- New sphere = sphere + cylinder. Add them algebraically (watch the signs).
- Flip the cylinder sign. −1.00 becomes +1.00; +0.50 becomes −0.50. The amount stays the same.
- Rotate the axis 90°. If the axis is 90 or less, add 90; if it's more than 90, subtract 90. Axes are written from 1 to 180.
Worked example
Transpose +2.00 −1.00 × 180:
- Sphere: +2.00 + (−1.00) = +1.00
- Cylinder: −1.00 → +1.00
- Axis: 180 − 90 = 090
+2.00 −1.00 × 180 = +1.00 +1.00 × 090
More examples
| Written as | Transposed |
|---|---|
+2.00 −1.00 × 180 | +1.00 +1.00 × 090 |
−1.50 +0.50 × 045 | −1.00 −0.50 × 135 |
plano −1.25 × 090 | −1.25 +1.25 × 180 |
+0.50 −1.50 × 170 | −1.00 +1.50 × 080 |
Check your work with a power cross
A power cross shows the power in each principal meridian. The meridian at the cylinder axis has only the sphere power; the meridian 90° away has sphere + cylinder. For +2.00 −1.00 × 180:
| Meridian | Power |
|---|---|
| 180 | +2.00 D |
| 090 | +1.00 D |
Do the same for +1.00 +1.00 × 090 and you get the identical cross: +1.00 at 090 and +2.00 at 180. Same lens, same powers — that's how you know the transposition is right.
Spherical equivalent
SE = sphere + (cylinder ÷ 2)
For +2.00 −1.00 × 180, SE = +2.00 + (−0.50) = +1.50 D. The transposed form gives the same answer: +1.00 + (+0.50) = +1.50 D.
Common mistakes
- Forgetting to rotate the axis (or rotating it to 270 instead of 090).
- Changing the cylinder amount instead of only its sign.
- Sign errors when adding the sphere and cylinder — write the signs out every time.
Frequently asked questions
What does it mean to transpose a prescription?
Transposing rewrites a sphero-cylindrical prescription from plus-cylinder form to minus-cylinder form (or the reverse). The lens itself does not change; only the way it is written does.
What are the steps to transpose a prescription?
Add the sphere and cylinder to get the new sphere, change the sign of the cylinder, and rotate the axis by 90 degrees, keeping it between 1 and 180.
What is +2.00 −1.00 × 180 in plus-cylinder form?
+1.00 +1.00 × 090. The new sphere is +2.00 + (−1.00) = +1.00, the cylinder becomes +1.00, and the axis rotates from 180 to 090.
Does transposition change the spherical equivalent?
No. Because both forms describe the same lens, the spherical equivalent (sphere + half the cylinder) is identical before and after transposition.