Refractive Focus

Vector Representation of an Optical Correction

Refractive Focus

Vector Representation of an Optical Correction


Several notation methods are commonly utilized by eyecare professionals and researchers to report the prescription incorporated into an ophthalmic correction. Positive cylinder and negative cylinder notations are most commonly utilized in ophthalmology and optometry, respectively. These notations directly communicate the combination of spherical and cylindrical lenses (including the cylindrical lenses’ orientation) that corrects a patient’s ametropia and directly relate to quantities under clinicians’ control with a phoropter or loose lens trial kit. Further, it is a straightforward operation to perform the calculations necessary to vertex corrections in both positive and negative cylinder forms between the spectacle and contact lens planes (a common clinical task).

But while convenient and appropriate for a number of uses, these notations do have limitations when more complex analyses of refractive error data are desired.

An Alternative—Power Vector Representation

In one attempt to address these limitations, an alternative representation, the power vector representation, was described in a review article by Thibos, Wheeler, and Horner (1997). This representation was put in the historical context of prior work by a number of investigators.

Said succinctly, the power vector method places all aspects of the refraction on a set of isomorphic (same units: diopters) orthogonal axes. This results in the abandonment of the polar coordinate system (r,θ) used to specify cylinder lens power and axis (a component of both positive and negative cylinder notations) in favor of representing cylinder in the Cartesian coordinate system (x,y). Sphere (S), cylinder (C), and axis (A) are transformed into a spherical lens of power M and two Jackson cross cylinders, one at axis 0° with power J0 and the other at axis 45° with power J45. This allows the entire prescription to be represented by a single vector in three-dimensional space. Figure 1 shows an example refraction represented using both negative cylinder (left) and power vector (right) notations.

Figure 1. Power vector notation takes the traditional sphere (S) cylinder (C) and axis (A) notation and, as described by Thibos et al (1997), transforms these variables into a spherical lens of power M and two Jackson cross cylinders, one at axis 0° with power J0 and the other at axis 45° with power J45.

The Fundamental Benefits of the Power Vector Notation

Why is such a method necessary? Thibos et al (1997) report on the rationale in their work as follows: “One benefit of our approach is that basic optometric data in the form of refractive errors and prescriptive lenses become amenable to statistical analysis, and are readily visualized graphically using a frame of reference, which is based on familiar optometric quantities.”

The mathematics of the method are surprisingly straightforward (please refer to the original work for those details), which facilitate the adoption of this method in vision research. One proxy for evaluating the interest, uptake, and impact of the method is the number of times the paper has been cited by others in the scientific literature (408 as of this writing) (Scopus search, 2014). The referencing literature shows that the method has been applied in attempts to, for example, quantify distributions of refractive error within a population, calculate a change in refraction, calculate an “average refraction” from a series of measurements obtained with an autorefractor, calculate the agreement between methods of obtaining a refraction, etc.

While facilitating statistical analysis, one notable inconvenience of the power vector representation is that the values representing the refraction no longer directly relate to quantities under clinicians’ control with a phoropter or loose lens trial kit and require an additional transformation back to the familiar (S C A) format for that purpose. Further, Thibos et al (1997) state in their development of the power vector method that it applies only to thin lenses and therefore lacks the generality of some alternate methods.

An Example from the Literature

In hopes of highlighting the utility of the power vector method, one instance in which the method was used to answer a specific question related to refraction is highlighted here.

“What is the repeatability of subjective refraction in myopic and keratoconic patients?” Raasch et al (2001) utilized the power vector method to examine the differences in two subjective refractions separated in time performed on keratoconic eyes as well as on myopic eyes. As stated by Raasch et al (2001), using the power vector method allowed the investigators to “express changes in dioptric power with one number, regardless of whether the change resulted from a change in sphere power, cylinder power, or cylinder axis or any combination of the three.”

Figure 2 puts the words of the investigators into pictorial form: the difference between these two example refractions was reduced to a single quantity: the distance between two vectors representing the individual refractions. The result of this work is an increased understanding of the variability in subjective refraction on two populations. One result: the investigators reported that the difference between test and retest of manifest (naked eye) refraction of keratoconus subjects who typically wear rigid contact lenses is roughly five times that seen in manifest refraction of myopes (1.28D versus 0.20D, respectively).

Figure 2. An example demonstrating the use of the power vector method to determine the difference between two different refractions.


In my opinion, there is no pressing reason for the power vector method to take the place of the more common plus and minus cylinder forms for many clinical applications. However, the utility of the power vector method when answering questions related to refraction is apparent. Further, a working knowledge of power vectors is useful in interpreting the refraction literature. CLS

For references, please visit and click on document #225.

Dr. Marsack completed a PhD in Physiological Optics and Vision Science at The University of Houston, College of Optometry. His research interests include optical aberration of the eye, custom and pseudo-custom correction of optical aberration, visual performance, metrics predictive of visual performance, and ocular drug delivery.