Rx Transposition
Convert minus-cylinder to plus-cylinder form; axis rotates 90°.
About this calculator
Eyeglass and contact lens prescriptions for astigmatism can be written in either of two mathematically equivalent notations: minus-cylinder form (the convention most ophthalmologists and optometrists in the US use) or plus-cylinder form (common among some lens labs and older prescriptions). Converting between them -- called transposition -- uses three rules applied together: the new sphere power equals the old sphere power plus the old cylinder power; the new cylinder power flips sign (positive becomes negative or vice versa) while keeping the same magnitude; and the new axis shifts by exactly 90 degrees, computed to stay within the standard 0-180 degree convention (adding 90 if the original axis is 90 or below, subtracting 90 if it is above 90).
All three changes together describe the exact same physical lens -- transposition does not change what the lens actually corrects, only how the same correction is written down. This calculator converts a single sphere/cylinder/axis entry -- if converting a full prescription, each eye (OD/right, OS/left) has its own independent sphere, cylinder, and axis values, so each eye needs to be entered and converted separately.
Medical Disclaimer
This calculator is for informational and educational purposes only. It is not a substitute for professional medical advice, diagnosis, or treatment. Always consult a qualified healthcare provider before making decisions about your health. Never disregard professional medical advice or delay seeking it because of results from this tool.
Inputs
Results
Sph (−cyl)
-2.5
How to Use This Calculator
- Locate your eyeglass prescription from your optometrist or ophthalmologist.
- Enter the Sphere (D), Cylinder (D), and Axis (°) for one eye (OD or OS).
- Read the transposed Sph/Cyl/New axis values in the output panel — the same lens written in the other notation.
- Repeat with the other eye's sphere, cylinder, and axis values if you need both eyes converted.
- Compare the output with your lens order form, and consult your eye-care professional before ordering lenses.
How the result changes with Sphere (D)
| Sphere (D) | Sph (−cyl) |
|---|---|
| -6.25 | -6.25 |
| -3.75 | -3.75 |
| -1.88 | -1.88 |
| -1.25 | -1.25 |
What each input means
- Sphere (D)
- The spherical power from your prescription in minus-cylinder form. Negative values correct nearsightedness.
- Cylinder (D)
- The cylinder power for astigmatism correction. In minus-cylinder form, this is typically negative.
- Axis (°)
- The cylinder axis in degrees (1-180). During transposition, the axis rotates by 90 degrees.
What each result means
- Sph (−cyl)
- Sphere value in minus-cylinder notation (your original input).
- Cyl (−cyl)
- Cylinder value in minus-cylinder notation.
- Sph (+cyl)
- Sphere value after transposing to plus-cylinder notation.
- Cyl (+cyl)
- Cylinder value in plus-cylinder notation (sign is flipped).
- New axis
- The transposed axis after rotating 90 degrees.
How this is calculated
Worked example, using the default values
- Identify Input ParametersSphere (D) = -2.5, Cylinder (D) = -0.75, Axis (°) = 180 = 3 input(s) provided
- Calculate SphSph = sph-2.5 = -2.5
- Calculate CylCyl = cyl-0.75 = -0.75
- Calculate SphSph = round((sph + cyl) * 100) / 100-3.25 = -3.25
Engine last updated . Checked against 3 independently-derived tests — how we verify calculators. Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.
Frequently Asked Questions
Why does the cylinder value simply flip sign during transposition?
Minus-cylinder and plus-cylinder notation describe the same astigmatism correction from two different reference axes, and flipping the cylinder's sign while adding it to the sphere (new sphere = old sphere + old cylinder) is the algebraic adjustment that keeps the total correction along both principal meridians identical. The magnitude of the cylinder power never changes -- only its sign and which axis it is referenced from.
Why does the axis rotate by exactly 90 degrees during transposition, not some other amount?
The two principal meridians of an astigmatic lens are always 90 degrees apart by definition. Transposition swaps which of the two meridians the cylinder power is referenced to, and swapping between two meridians that are 90 degrees apart means the stated axis must also shift by exactly 90 degrees to keep describing the same physical lens.
Does converting a prescription to the other notation change what the lens corrects?
No -- minus-cylinder and plus-cylinder notation are two different ways of writing down the identical physical lens. Transposition is purely a notational conversion: the sphere, cylinder, and axis all change together in a way that exactly cancels out, so the lens ground from either notation corrects vision identically.
Do both eyes use the same sphere, cylinder, and axis values?
No -- a full prescription lists separate sphere, cylinder, and axis values for the right eye (OD) and left eye (OS), since most people have somewhat different refractive errors in each eye. This calculator converts one eye's values at a time, so enter and convert the right eye's numbers, then repeat for the left eye's numbers.
Related Calculators
The questions that sit next to this one — chosen by subject, including calculators filed under a different category.
Glasses to Contacts Rx Calculator
Convert your glasses prescription to contact lens prescription using vertex distance adjustment for powers above ±4D.
Music & AudioKey Transposition Calculator
Transpose music from one key to another. Find the target key, interval name, capo position, and common transposition uses.
Math & StatisticsStandard Form Calculator
Convert numbers to standard form (scientific notation). Express large or small numbers in a×10ⁿ format.
OptometryLens Thickness Calculator
Estimate center and edge thickness of eyeglass lenses based on prescription power, lens index, and frame size.
Math & StatisticsScientific Notation Calculator
Convert numbers to scientific notation and engineering notation. Control significant figures and see coefficient and exponent.
More in Medical & Clinical.