This calculator combines two cylindrical lenses at different axes into a sphero-cylindrical lens. Enter the power and axis of each cylindrical lens to calculate the resulting prescription.
A crossed cylinder is the result of combining two cylindrical lenses with their axes perpendicular to each other. This concept is fundamental in optometry and ophthalmology for:
Ophthalmic prescriptions are typically written in sphero-cylindrical form:
S D Sph × C D Cyl @ A°
Where:
Example: +2.00 -1.50 × 45°
The calculation of crossed cylinders uses power vector analysis, which represents cylindrical lenses as vectors. This method was developed by Larry Thibos and involves:
J₀ = -C/2 × cos(2A)
J₄₅ = -C/2 × sin(2A)
Where:
To combine two cylinders:
C = -2 × √(J₀² + J₄₅²)
A = 0.5 × arctan(J₄₅/J₀) + K
Where K = 0° if J₀ > 0, K = 90° if J₀ < 0, and K = 0° if J₀ = 0 and J₄₅ ≥ 0, or K = 90° if J₀ = 0 and J₄₅ < 0.
Consider combining a -2.00 D cylinder at 30° with a -1.50 D cylinder at 120°:
The resulting lens would be 0.00 -0.50 × 30°