Article Date: 4/1/2012 |

To refine a toric soft contact lens prescription, we typically perform a spherocylindrical overrefraction over the original lens and then calculate the product of the two crossed cylinders. Fortunately, the complicated equations required for this calculation have been programmed into several easy-to-use calculators. Although they are convenient tools, these calculators do not allow us to visualize the interaction between crossed cylinders. I believe using vector analysis to perform these calculations gives us a greater appreciation of the mechanics involved. The method I propose requires converting cylinder power into a vector. Doing this shows us how an overrefraction shifts the cylinder and axis of a lens to refine the prescription.

A vector represents a physical quantity, such as a push or a pull, that has a specific direction and magnitude. By performing vector analysis, we break down these forces into basic components to understand how they interact. If we apply the same principles to optics and describe cylinder power as a vector, we can simplify the process of measuring crossed cylinders and gain an enhanced understanding of how a cylindrical overrefraction affects a prescription. Vector analysis can teach us how to incorporate an overrefraction into the final correction.

A vector is represented pictorially by a line with an arrowhead pointing in the same direction as the force. Its length is proportional to its magnitude on some agreed scale.

Figure 1 (see page 44) shows a three-against-one tug-of-war, which is represented by the vectors below it. The light-blue arrow points left to represent the magnitude and direction of the single warrior, and the, dark-blue arrow (three times the length of the light blue arrow) points right to represent the magnitude and direction of the three opponents. The sum of the two vectors creates the net effect indicated by the black arrow (two times the length of the light blue arrow) pointing toward the right. Similarly, dioptric cylinder has a magnitude and a direction, CYL and AXIS respectively, and it can be expressed graphically as a vector.

**Figure 1: The black arrow shows the net effect of two vectors.**

In the physical world, opposing forces act 180° apart, but in optics, opposing cylinders act 90° apart. In the physical world, two-dimensional forces can act in any direction along 360°, but in optics, cylinder is limited to 180°. Thus, the opposite to 0° is not 180°, it is 90°. As we know, a –1.00D x 90 lens combined with a –1.00D x 180 cyl lens will cancel the cylinder to 0, creating a –1.00D spherical lens. To compensate for this and get a true graphic representation, we need to “stretch” 180° to a full circle, which causes the angles of cylindrical vectors to appear doubled. You can use Figure 2 (above) to plot cylindrical vectors.

**Figure 2: This grid can be used to plot cylindrical vectors.**

Figure 3 (below) shows an example of how to apply vector analysis to clinical practice. In this example, one ring equals –0.50DC. The vector (cylinder power) of the diagnostic lens (–2.00D x 30) is represented by the red arrow. The vector of the overrefraction (–1.00D x 45) is plotted as a blue arrow of proportional length pointed up and parallel to the 45° meridian. By placing the vector of the overrefraction at the head of the red arrow (heel to toe), we can add the vectors graphically. We form a triangle of vectors by drawing an arrow that connects the origin to the tip of the overrefraction vector, as shown by the green arrow. This vector is the result of the combination of the crossed cylinders. Its length (–2.87) and angle (36°) correspond to the cylinder and axis that should provide optimum vision. This graphical demonstration of the effect of the overrefraction on the lens prescription indicates that the next logical lens choice has a cylinder power of –2.75D x 40.

**Figure 3: This figure provides an example of how to apply vector analysis to clinical practice.**

Note the following caveats related to residual sphere and lens rotation.

** Residual sphere:** The total dioptric power of the lens plus the overrefraction must remain constant. Any residual loss or gain in cylinder power must be balanced by a corresponding loss or gain in sphere power. This residual is equal to the spherical equivalent (half the value) of the difference between the new cylinder and the sum of the two crossed cylinders. In this example: [(–2.00DC) + (–1.00DC)] – (–2.75DC) = (–0.25 DC)/2 = –0.12DS.

** Lens rotation:** Lens rotation also affects the incorporation of the overrefraction. This difference is negligible for minor amounts of lens rotation and low cylinder power of the overrefraction. Compensation is simple: Rotate the vector of the overrefraction in the opposite direction of the lens rotation. In the example, if the diagnostic lens rotates 10° counterclockwise, draw the vector of the overrefraction at 35° instead of 45° (a 10° clockwise rotation).

Seeing how crossed cylinders interact can aid prescribing in ways optical calculators cannot. Vector analysis lets you see how the overrefraction shifts the cylinder and axis. This makes correcting for astigmatism an intuitive process that you can accomplish fast in the exam room. With a simple sketch, you can easily quantify and apply adjustments. By describing cylinder power as a vector, you can graphically analyze the interaction between the lens and the overrefraction to better understand how to modify the prescription and choose the appropriate lens. **OM**

Dr. Kreda practices in a primary care setting in Lauderhill, Fla. He's a frequent lecturer and author. E-mail him at EyeRx@comcast.net. Or, send comments to optometricmanagement@gmail.com. |

*Optometric Management*, Volume: 47 , Issue: April 2012, page(s): 44 - 46 53