Transposition Calculator: Plus Cylinder to Minus Cylinder
This calculator transposes a sphero-cylindrical prescription between plus-cylinder and minus-cylinder form. Enter the sphere, cylinder and axis, and it returns the same lens written the other way. Transposition never changes the lens, only how its powers are written.
RX Transpose
Convert a prescription between plus-cylinder and minus-cylinder notation.
Formula
new sphere = sphere + cyl ; new cyl = −cyl ; new axis = (axis + 90) mod 180.
The three-step transposition rule
1. New sphere = sphere + cylinder, added algebraically with their signs.
2. New cylinder = same amount, opposite sign.
3. New axis = axis ± 90°, kept between 1 and 180. The calculator adds 90 and subtracts 180 if the result passes 180, so 90 becomes 180, never 0.
Worked examples
Plus cylinder to minus cylinder: -0.75 +1.25 × 170. New sphere: -0.75 + 1.25 = +0.50. Cylinder: -1.25. Axis: 170 + 90 = 260, minus 180 = 80. The calculator shows +0.50 -1.25 × 80, written × 080 on a prescription.
Minus to plus cylinder: +1.75 -0.75 × 35 becomes +1.00 +0.75 × 125.
Mixed astigmatism: +1.00 -3.00 × 180 becomes -2.00 +3.00 × 90. The sphere changes sign, which is correct: one meridian is +1.00 D and the other is -2.00 D.
Check your answer with the meridian powers
Both forms describe the same two principal meridians. In -0.75 +1.25 × 170, the 170° meridian has -0.75 D and the 80° meridian has -0.75 + 1.25 = +0.50 D. In +0.50 -1.25 × 80, the 80° meridian has +0.50 D and the 170° meridian has +0.50 - 1.25 = -0.75 D. If the meridian powers match, the transposition is right.
When to use it and common mistakes
Optometrists usually write minus cylinder and many ophthalmologists write plus cylinder, so opticians transpose to compare a new prescription with old glasses, to match a lensmeter reading taken the other way, and to order toric soft contact lenses, most of which are listed in minus cylinder.
Common mistakes: subtracting when the signs differ (-0.75 + 1.25 is +0.50, not -2.00); changing the sign of the sphere instead of the cylinder; writing an axis of 0 or 270 instead of 180 or 90; and leaving the axis unchanged. A plano sphere shows as +0.00 in the calculator; write it as plano (pl).
Transposition on the ABO exam
Transposition is a building block for other ABO problems: naming the type of astigmatism, finding the power in a meridian for Prentice's rule, and comparing prescriptions written in different forms. Check every answer against the mistakes above before you choose it. The spherical equivalent is a quick cross-check because it does not change on transposition. Review Prescription (Rx) Interpretation, then try the free ABO practice test.
Transposition examples: plus cylinder and minus cylinder
Transposition examples, as the calculator returns them (+0.00 means plano). The meridian column lists the power in each principal meridian, which is the same in both forms.
| Prescription entered | Transposed (calculator output) | Power in each meridian |
|---|---|---|
| -0.75 +1.25 × 170 | +0.50 -1.25 × 80 | +0.50 D at 80°; -0.75 D at 170° |
| +1.75 -0.75 × 35 | +1.00 +0.75 × 125 | +1.75 D at 35°; +1.00 D at 125° |
| -3.00 +2.00 × 90 | -1.00 -2.00 × 180 | -3.00 D at 90°; -1.00 D at 180° |
| +1.00 -3.00 × 180 | -2.00 +3.00 × 90 | -2.00 D at 90°; +1.00 D at 180° |
| +0.00 +1.50 × 15 | +1.50 -1.50 × 105 | +0.00 D at 15°; +1.50 D at 105° |
| +1.25 -1.25 × 100 | +0.00 +1.25 × 10 | +0.00 D at 10°; +1.25 D at 100° |
Frequently asked questions
How do you convert plus cylinder to minus cylinder?
Combine the sphere and cylinder algebraically for the new sphere, flip the cylinder's sign, and turn the axis 90°. For example, -0.75 +1.25 × 170 becomes +0.50 -1.25 × 080.
Does transposition change the prescription?
No. Both forms describe the same lens with the same power in each principal meridian. Only the notation changes, so the wearer sees exactly the same through either.
Why does an axis of 90 transpose to 180 and not 0?
Cylinder axes are written from 1° to 180°. The horizontal meridian is called 180 by convention, never 0. So 90 transposes to 180, and 180 transposes to 90.
Does the spherical equivalent change after transposition?
No. -1.00 -2.00 × 090 and its transposed form -3.00 +2.00 × 180 both have a spherical equivalent of -2.00 D. If your two forms give different spherical equivalents, recheck the transposition.
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