Plus and Minus Cylinder for the COA®: Transposition and Lensometry

Plus and minus cylinder are two ways to write one lens. To transpose, add the sphere and cylinder to get the new sphere, flip the cylinder’s sign, and turn the axis 90°. IJCAHPO lists transposing cylinder among the COA optics tasks. On a manual lensometer, the direction you turn for the second lines sets the form.

Last checked Nov 2, 2026. Confirm fees and dates with the certifying body. optical.education is independent and is not affiliated with or endorsed by IJCAHPO. Disclaimer.

Same lens, two notations

A spherocylinder has two principal meridians. Plus-cylinder and minus-cylinder notation are two routes to the same pair of meridian powers, so neither changes what the patient gets. Conventions differ from office to office, so you need to read and convert both. The Core Criteria Handbook lists transposing cylinder among the COA optics tasks (checked Nov 2, 2026).

The 3-step transposition

1. New sphere: add the cylinder to the sphere, keeping signs.
2. New cylinder: same amount, opposite sign.
3. New axis: for an axis of 90 or below, add 90°; above 90, take off 90°. The result always stays between 1 and 180.

A quick check: work out the two meridian powers before and after. If they match, the transposition is right.

Check with the calculator

Enter any lens in the transposition calculator to see the other form. Do the examples below by hand first. The lens transposition guide covers the topic for ABO candidates.

Plus or minus cylinder on the lensometer

On a manual lensometer, the single lines give the sphere. From there, the way you turn the power wheel for the triple lines decides the form: toward minus gives minus cylinder, toward plus gives plus cylinder. If the lines will not clear in the direction you want, rotate the axis 90° and start again from the single lines. Practice both on the lensometer simulator, and see lensometry for the COA for the full routine.

Worked examples

1. Plus to minus

−1.00 +2.00 × 080

1. Sphere: −1.00 + 2.00 = +1.00.
2. Cylinder: −2.00.
3. Axis: 080 + 90 = 170.

Result: +1.00 −2.00 × 170. Check: meridian 80 is −1.00 in both forms (+1.00 − 2.00), and meridian 170 is +1.00.

2. Minus to plus

+3.25 −0.75 × 145

1. Sphere: +3.25 − 0.75 = +2.50.
2. Cylinder: +0.75.
3. Axis: 145 − 90 = 055.

Result: +2.50 +0.75 × 055.

3. A sphere that shifts by the full cylinder amount

−4.50 +1.25 × 010 becomes −4.50 + 1.25 = −3.25, cylinder −1.25, axis 100: −3.25 −1.25 × 100.

Practice set

Original problems. Transpose each one, then check it with the calculator.

1. +0.50 +1.50 × 120 to minus cylinder.

   Show answer

   Sphere +0.50 + 1.50 = +2.00; cylinder −1.50; axis 120 − 90 = 030. +2.00 −1.50 × 030

2. −2.75 −2.25 × 065 to plus cylinder.

   Show answer

   Sphere −2.75 + (−2.25) = −5.00; cylinder +2.25; axis 065 + 90 = 155. −5.00 +2.25 × 155

3. plano +1.00 × 090 to minus cylinder.

   Show answer

   Sphere 0 + 1.00 = +1.00; cylinder −1.00; axis 090 + 90 = 180. +1.00 −1.00 × 180

4. +1.75 −1.75 × 045 to plus cylinder.

   Show answer

   Sphere +1.75 − 1.75 = 0; cylinder +1.75; axis 045 + 90 = 135. plano +1.75 × 135

5. On a manual lensometer the single lines clear at −1.00 at axis 30, and the triple lines clear at +0.50 turning toward plus. Write the lens both ways.

   Show answer

   Plus form: −1.00 +1.50 × 030. Minus form: sphere −1.00 + 1.50 = +0.50, cylinder −1.50, axis 120. −1.00 +1.50 × 030 = +0.50 −1.50 × 120

Standard texts used for this page: Ledford JK. Certified Ophthalmic Assistant Exam Review Manual, 3rd ed. SLACK, ch. 5 (reading plus and minus cylinder) and ch. 15 (transposition); Brooks CW, Borish IM. System for Ophthalmic Dispensing, 3rd ed. Butterworth-Heinemann, ch. 6; and the University of Iowa Optics Review, linked under sources.

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