Combine two sphero-cylindrical lenses whose cylinder axes differ (obliquely crossed cylinders), or take one away from the other: glasses plus an over-refraction, or refraction minus corneal astigmatism. Results in minus and plus cylinder form, with the power-vector working. For opticians and optometrists.
Worked examples
Two plano +1.00 cylinders, at 0° and 45°. The classic formulas give a resultant cylinder of √(1² + 1² + 2 × 1 × 1 × cos 90°) = √2 = 1.41 D, an axis where tan 2α = sin 90° ÷ (1 + cos 90°) = 1, so α = 22.5°, and a sphere of (1 + 1 − 1.41) ÷ 2 = 0.29 D: +0.25 +1.50 × 23, which is +1.75 −1.50 × 113 in minus cylinder (exactly +1.71 −1.41 × 112.5).
Glasses plus an over-refraction. −2.00 −1.00 × 180 worn, over-refraction +0.25 −0.50 × 45: the new prescription is −2.00 −1.00 × 13.
Residual astigmatism. Refraction −2.00 −1.50 × 180, corneal astigmatism −1.00 × 170 (as a lens): what a spherical rigid lens would leave is −2.00 −0.75 × 16.
Formulas
- Power vectors:
M = S + C/2, J0 = −(C/2)·cos 2α, J45 = −(C/2)·sin 2α. - Back, in minus cylinder:
J = √(J0² + J45²), C = −2J, S = M + J, α = ½·atan2(J45, J0). - Classic form for cylinders C₁ × 0 and C₂ × θ:
C = √(C₁² + C₂² + 2C₁C₂·cos 2θ), tan 2α = C₂·sin 2θ ÷ (C₁ + C₂·cos 2θ), added sphere (C₁ + C₂ − C) ÷ 2. The tests check both methods agree on 10,000 random lens pairs.
Other optics tools: vertex distance, lens thickness and lens tilt and wrap.
References
- Thibos LN, Wheeler W, Horner D. Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error. Optom Vis Sci 1997;74(6):367–375.
Formulas checked on 19 October 2026. Thin lenses in contact; for eye-care professionals.
Frequently asked questions
Why can't two cylinders just be added?
A cylinder has a power and a direction (its axis), so two cylinders at different axes combine like vectors, not like numbers. Only when the axes are the same, or 90° apart, do the powers add simply. Between those, the result is a cylinder of a new power at an axis somewhere between the two, plus some sphere.
How does the calculator combine them?
Each lens is turned into power vectors, M = S + C/2, J0 = −(C/2)·cos 2α and J45 = −(C/2)·sin 2α (Thibos, Wheeler and Horner, 1997). The vectors of thin lenses in contact add up; the total is turned back into sphere, cylinder and axis. It gives exactly the same answer as the classic obliquely-crossed-cylinder formulas, and the working is shown under the result.
What is an over-refraction?
A refraction done over the patient's current glasses or contact lenses. Adding the over-refraction to the lens being worn gives the full new prescription; if the over-refraction has a cylinder at a different axis, the two have to be combined as crossed cylinders, which is what this calculator does.
What is residual astigmatism?
The astigmatism left when a spherical (rigid) contact lens neutralises the cornea's astigmatism: roughly the refractive astigmatism minus the corneal astigmatism. Choose “Take the second lens away” and enter the corneal astigmatism, written as a cylinder, as the second lens.
Minus or plus cylinder?
Both describe the same lens: the plus-cylinder form is the transposition of the minus-cylinder form (add the cylinder to the sphere, change the cylinder's sign, turn the axis by 90°). The result is shown both ways, rounded to 0.25 D and the whole degree, and exactly.
