Transposing an eyeglass prescription converts the numerical values from one format to another without changing the lens’s overall optical power. This conversion is necessary because prescriptions that include astigmatism correction can be written in either a plus (+) cylinder or a minus (-) cylinder notation. Depending on the country, the optometrist’s preference, or the specific optical laboratory’s equipment, one form of notation is often required over the other for lens fabrication.
Components of an Eyeglass Prescription
A standard eyeglass prescription contains three main values that define the lens power for each eye: Sphere (SPH), Cylinder (CYL), and Axis.
The SPH value, measured in diopters, indicates the amount of lens power needed to correct for nearsightedness or farsightedness. A minus sign denotes nearsightedness, while a plus sign signifies farsightedness, determining the overall focusing power of the lens.
The CYL value, also measured in diopters, represents the power required to correct astigmatism, a common condition where the eye’s cornea or lens has an irregular, non-spherical shape. This cylindrical power is what corrects the distortion or blurriness caused by the astigmatism. If a prescription shows no value for CYL, it means there is no astigmatism correction needed.
The third component is the Axis, which is an angle value between 1 and 180 degrees that specifies the exact orientation of the cylindrical correction on the lens. Astigmatism correction must be placed at a precise angle to align with the irregular curve of the eye.
The Transposition Method
Transposing a prescription requires a specific three-step mathematical rule to ensure the new notation is optically identical to the original. This method is used for prescriptions with a spherical and cylindrical component.
The first step involves calculating the new Sphere (SPH) value by algebraically adding the original Sphere power and the original Cylinder power.
The second step is to determine the new Cylinder (CYL) value, which is done by simply changing the sign of the original Cylinder power while keeping the magnitude the same. For instance, a +2.00 cylinder becomes a -2.00 cylinder, and a -1.50 cylinder becomes a +1.50 cylinder.
The third step requires adjusting the Axis by 90 degrees. If the original Axis value is 90 degrees or less, you must add 90 degrees to it. Conversely, if the original Axis value is more than 90 degrees, you must subtract 90 degrees to keep the new axis within the standard range of 1 to 180 degrees.
Applying the Formula
The transposition method ensures that the final lens still provides the correct power at every meridian, even though the numbers look different.
Consider an original prescription of -2.00 -1.50 x 180, which is in minus-cylinder form. Applying the formula yields a new Sphere of (-2.00) + (-1.50) = -3.50; the new Cylinder is +1.50 (the opposite sign); and the new Axis is 180 – 90 = 90. The transposed prescription is therefore -3.50 +1.50 x 90.
For a prescription written in plus-cylinder form, such as +1.00 +2.50 x 045, the steps are similar. The new Sphere is calculated as (+1.00) + (+2.50) = +3.50; the new Cylinder becomes -2.50; and the new Axis is 45 + 90 = 135. This results in a transposed prescription of +3.50 -2.50 x 135.
A complex example involving both plus and minus sphere values is +2.50 -2.00 x 105. The new Sphere is calculated by adding the original sphere and cylinder algebraically: (+2.50) + (-2.00) = +0.50. The new Cylinder is the original cylinder with the sign reversed, becoming +2.00. Since the original Axis is 105, which is greater than 90, we subtract 90, resulting in a new Axis of 15. The final transposed prescription is +0.50 +2.00 x 15.