Transposition rewrites a sphero-cylindrical prescription from plus-cylinder to minus-cylinder form, or back, without changing the lens at all. The two forms describe the same optical correction; only the notation differs. Enter sphere, cylinder and axis, and this calculator returns the transposed prescription instantly, along with the spherical equivalent that both forms share.
The Three Steps
Transposition follows the same three steps every time:
- New sphere = add the sphere and cylinder together.
- New cylinder = keep the same number, reverse the sign.
- New axis = rotate 90 degrees: add 90 if the axis is 90 or less, subtract 90 if it is over 90.
New sphere = sphere + cylinder. New cylinder = same power, opposite sign. New axis = axis ± 90 degrees.
Worked Example
Take +1.00 +1.00 × 090 in plus-cylinder form:
- New sphere: +1.00 + 1.00 = +2.00
- New cylinder: +1.00 becomes −1.00
- New axis: 90 + 90 = 180
The minus-cylinder form is +2.00 −1.00 × 180. Both prescriptions bend light identically, and a lensmeter reads them the same. A second case, −2.25 +0.75 × 135, transposes to −1.50 −0.75 × 045: sphere −2.25 + 0.75 = −1.50, cylinder reversed to −0.75, axis 135 − 90 = 45.
Why Both Forms Exist
Most optometric phoropters are set up in minus-cylinder form, while some ophthalmic practice and older equipment work in plus-cylinder. Labs and dispensing software may expect one form. Reading a prescription written in the unfamiliar form, or ordering in the wrong one, is where transposition errors creep in. Because the optical power is unchanged, the spherical equivalent (sphere plus half the cylinder) is identical in both forms, which is a quick way to check that a transposition was done correctly. For what the cylinder and axis describe, see astigmatism explained.
A Note on Axis
The axis always stays between 1 and 180 degrees. Adding or subtracting 90 keeps it in range: an axis of 180 transposes to 90, and an axis of 30 transposes to 120. The cylinder rotates exactly 90 degrees because the two principal meridians of the lens are perpendicular.
Frequently Asked Questions
How do you transpose a prescription?
Add the sphere and cylinder for the new sphere, reverse the sign of the cylinder, and rotate the axis 90 degrees (add 90 if it is 90 or less, subtract 90 if it is over 90). The result is the same lens written the other way.
Does transposition change the prescription?
No. Plus-cylinder and minus-cylinder forms describe the identical lens. The spherical equivalent (sphere plus half the cylinder) is the same in both, which is a quick way to check your work.
Why do optometrists and ophthalmologists write cylinder differently?
Most optometric phoropters are set in minus-cylinder form, while some ophthalmic practice and older equipment use plus-cylinder. Both are valid; transposition converts between them.
What happens to the axis when you transpose?
It rotates exactly 90 degrees and stays between 1 and 180. An axis of 180 becomes 90; an axis of 45 becomes 135.
Can transposition produce a plano sphere?
Yes. If the sphere and cylinder are equal and opposite, the transposed sphere is zero (plano). The arithmetic is the same; only the written form looks different.
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