Crossed Cylinders Calculator

The method: add the cylinders as power vectors

A cylinder cannot be added to another at a different axis by simple arithmetic. Split each one into a spherical part and two astigmatic parts, add those, and rebuild:

Worked example: over-refraction

A wearer's glasses are plano −2.00 × 180. Over them, −0.50 × 30 gives the best vision. Worked by hand:

Cylinder = −2 × √(1.125² + 0.217²) = −2.29. Sphere = −1.25 + 1.146 = −0.10. Axis = ½ × arctan(0.217 ÷ 1.125) = 5°. The new prescription is −0.10 −2.29 × 5, ordered in quarter steps as plano −2.25 × 005.

The calculator gives its result in the same minus-cylinder form. For these inputs it shows -0.10 D, -2.29 D and 5°. In plus-cylinder form the same lens is −2.40 +2.29 × 95 (see the plus to minus cyl converter).

Two checks worth knowing

• Equal cylinders 90° apart make a sphere: −1.00 × 180 with −1.00 × 90 is −1.00 D with no cylinder. The calculator shows -1.00 D with a cylinder of 0.00 D and no axis.
• Cylinders at the same axis simply add: −1.00 × 180 with −0.50 × 180 is plano −1.50 × 180. The calculator shows 0.00 D, -1.50 D and 180°.

Where it is used

• Over-refraction over existing spectacles or a toric contact lens, as in the example.
• Working out the residual astigmatism when a toric contact lens settles at the wrong axis.

Confirm any combined prescription by refraction before it is ordered; see the medical disclaimer.

Questions