It is well known that spherocylindrical powers expressed in the usual form of sphere, cylinder, and axis may be represented in a three-dimensional space by a single point. In fact, there is not a unique representation, but rather several coordinate vector bases have been defined for thin lenses. ^{1–6} Due to its graphical nature, ^{7–10} the representation of dioptric powers in these vector spaces has demonstrated its usefulness, especially in scientific research applications, where numerical and statistical data analysis is needed. Recently, the extension of these concepts to more general systems (thick systems, for example) and the relationship between several bases and vectors have been provided. ^{11} Because astigmatic optical systems are better understood in terms of matrices and vectors, the dioptric power space representation has been used to revise several clinical optometric techniques. ^{12} However, to our knowledge, no attempt has been made to include some conventional subjective techniques into this three-dimensional analysis framework. Taking into account that these techniques are clinically easy to use but that the working principles of some of them are rather difficult to understand, it is expected that their analysis in the dioptric power space will provide to refractionists new insights into the clinical procedures that they regularly use.

The purpose of this paper is twofold: First we give a new heuristic derivation of a power vector representation in a three-dimensional space from the familiar sine-squared law for the power of a refracting surface. The final result, which coincides with the previous one developed, ^{3, 5} is derived in a very intuitive way. Second, it is shown that this representation can be used in the analysis of the optical principles of different subjective refraction procedures such as the stenopaic slit refraction, the Barnes astigmatic decomposition method, and the Jackson cross-cylinder (JCC) procedure. Furthermore, the analysis of the stenopaic slit refraction and the Barnes method in this framework leads us to propose some modifications that will improve the performance of both techniques.

#### THREE-DIMENSIONAL POWER SPACE

An astigmatic power specified by sphere, cylinder, and axis as S C×α can also be represented, within certain approximations, by a continuous function: that describes the power of a spherocylindrical surface along the meridian at an angle θ. ^{13} This function, which is widely known as meridional power, has also been called the curvital power ^{11} to distinguish it from the torsional power, ^{14} which is perpendicular to the meridian. By applying trigonometrical identities, Equation 1 can be expressed alternatively as; or

According to Equation 2, the power along the θ meridian can be separated in two parts: the spherical equivalent powerMATHand the “pure” cylinder

Equation 2 Image Tools |
Equation U4 Image Tools |

The form of Equation 5 suggests the use of a graphical representation of the astigmatic power that is essentially the same geometrical construction used for the addition of harmonic waves in physics. In effect, it can be represented in a plane as the combination of two orthogonal components corresponding to θ = 0° and θ = 45° (see Equation 3), as shown in Fig. 1. Formerly, Equation 5 represents a phasor of amplitude C/2 and phase (or phase shift) 2α. ^{15} Therefore, the specification of an astigmatic power can be done with the use of three parameters; namely, the components along the axes C_{0} and C_{45} of Fig. 1 and the spherical equivalent M, which corresponds to the dioptric value of the circle of least confusion (CLC). These components, can be used to define a uniform three-dimensional space where a single point represents the spherocylindrical power P_{θ} (Fig. 2). In this way we have presented an alternative derivation of the result obtained first by Deal and Toop ^{3} and later by Thibos et al. ^{5}

Equation 3 Image Tools |
Equation 5 Image Tools |
Figure 1 Image Tools |

Figure 2 Image Tools |

The vector space defined by Equation 6 is of special interest because it results in a very intuitive representation: as can be seen in Fig. 2, the spherical equivalent M of the spherocylindrical power is just along one axis, whereas the other two axes, C_{0} and C_{45}, define the plane of the Jackson cross-cylinder ^{3} (also called astigmatic plane ^{5}). Furthermore, the modulus of the vector (C_{0}, C_{45}, M), defined as can be used as a single scalar magnitude characterizing P_{θ}. This magnitude, derived also by Rabbetts ^{4} and Thibos et al., ^{6} was proposed by Raasch ^{16} as the optimum one to describe the relation between the blur created by astigmatic refractive errors and visual acuity. The smaller the vector, the better the visual acuity associated with it. From the values C_{0}, C_{45}, and M, the astigmatic power can be reobtained in the standard notation as ^{6}The result summarized by Equations 6 and 7 is an elegant and useful way to express the classical clinical representation of sphere, cylinder, and axis. Furthermore, this representation is very useful for understanding the optical principles of several refractive methods.

Equation 6 Image Tools |
Equation 7 Image Tools |

##### STENOPAIC SLIT REFRACTION

The subjective refraction performed with the stenopaic slit is a procedure especially useful in cases of high astigmatism and poor visual acuity. ^{17} The stenopaic slit placed in front of the eye isolates an ocular meridian on which to perform the refraction, independently of the other meridians. The combination of the slit with the refractive error modifies both the location and size of the CLC. In fact, if S/C × α is the ocular refractive error (correction) and β is an arbitrary orientation of the stenopaic slit in front of the eye, it can be proved that the resultant residual refractive error (R_{β}) in the three-dimensional space is a vector with the following components ^{18} :

This means that each orientation of the slit actually produces a different residual refractive error. The slit projects the vector representing the ocular refractive error E onto the plane that, containing the M axis, is perpendicular to the one defined by the slit orientation. Therefore, when a stenopaic slit is aligned with one of the principal meridians of the eye, the projection of E for these two orientations of the slit always lies on the M axis, and the resultant refractive error is purely spherical. For example, the profile that the tip of the vector R_{β} describes for a symmetric mixed astigmatism E = (+1 − 2 × 90°) (whose components (C_{0}, C_{45}, M) computed with Equation 6 6 are (−1, 0, 0)) when the slit scopes the ocular meridians within the range (0° to 180°) is represented in Fig. 3. As in this example, the CLC is at the retina, the substitution of Equation 9 into Equation 7 with M = 0 results in a vector R_{β} with constant modulus, i.e., for different orientations of the slit, the tip of the vector R_{β} runs on the surface of a sphere, following a curve in the direction showed by the arrows. For _{β} = 180° and β = 90°, the residual refractive error is a pure sphere whose dioptric power is one-half of the cylinder (±1 D in our example), with components (0, 0, −1) and (0, 0, 1), respectively, in the dioptric power space. On the other hand, for β = 45° and β = 135°, the residual refractive error coincides with the refractive error, i.e., R_{β} = E = (−1, 0, 0). This means that when the slit is placed 45° away from the principal meridians, it produces no effect on the location of the CLC, which remains at the retina. In fact, according to Equation 1 in this particular case, these orientations of the slit isolate the meridians of zero dioptric power.

Equation 9 Image Tools |
Figure 3 Image Tools |

To explain the stenopaic slit refraction procedure, let us restrict our analysis for simplicity to the previous particular example. The refractive error E can be considered as formed by two cylinders: E_{90} = −1 × 90° and E_{180} = +1 × 180° (Fig. 4 a), whose components are (−0.5, 0, −0.5) and (−0.5, 0, 0.5), respectively, both with C_{45} = 0°. Therefore, the analysis in the power space can be performed for this particular case in the MC_{0} plane (Fig. 4 b). We have chosen an example with the CLC at the retina because, actually, this is the starting point of the stenopaic slit procedure. The stenopaic slit is placed in front of the eye being tested, and it is slowly rotated while the patient’s attention is directed to the test chart. The patient is instructed to indicate which position of the stenopaic slit gives the best vision, so the meridian parallel to this orientation of the slit coincides with the axis of the correcting minus cylinder. This step can be graphically explained considering the residual refractive error R_{β} produced by the stenopaic slit (Fig. 3). In our example, when the slit is at 90° (Fig. 5 a), the CLC moves toward the horizontal focal line, behind the retina, in such a way that the refractive error is reduced to almost a focal point. In other words, the residual error is almost spherical, and its dioptric power coincides with the E_{180} component. As illustrated in Fig. 5 b, with the stenopaic slit at 90°, the vector components change from E = (−1, 0, 0) to R_{90} = (0, 0, 1) (Fig. 3). Because the resultant residual error has a positive value of M, in a conventional procedure, the eye accommodates (adding plus sphere power) to return the CLC to the retina (M = 0). This makes the refractive residual error vector length R_{90} diminish until it disappears, giving the patient the best vision. There is a unique orientation of the slit (90° in our example) that when combined with the accommodation, can correct the ocular refractive error; for the rest of the orientations (if M > 0), the residual refractive error is a spherocylinder. After finding the best vision meridian, the slit must be rotated 90° to locate the other principal meridian. In our example, this situation should be obtained with the slit at 180°. Then the CLC moves toward the vertical focal line in front of the retina (Fig. 6 a). Now the residual refractive error is almost a sphere whose dioptric power coincides with the one of the E_{90} component (Figs. 3 and 6 b). Therefore, the refractive power error has again been isolated, now along the 180° meridian. However, in this case, the refractive residual error R_{180} can not be reduced by use of accommodation. Therefore, in this position, the slit provides poorer vision than in the previous one.

Figure 4 Image Tools |
Figure 5 Image Tools |
Figure 6 Image Tools |

Once the principal meridians have been detected, the slit is returned to the clearest position, and the eye is fogged and unfogged with spheres until the best acuity is reached (Fig. 7 a). In our example, this sphere S has a power of +1 D, with components (0, 0, 1) (Fig. 5 b). With that lens and the slit in the vertical position, the residual error is reduced to zero, i.e., R_{90} − S = 0 (Fig. 7 b). Performing as before and removing the lens S, if the slit is placed at 180°, the spherical correcting power is now −1 D. In conclusion, by combining the two sphere powers, the +1 −2 × 90° correcting lens is obtained.

It must be taken into account that the procedure is not as precise as would be desirable because the near focal points that the slit produces at 90° and 180° are not quite superimposed on their respective focal lines (Figs. 5a, 6a, and 7 a). This justifies why the stenopaic slit refraction must be controlled by fogging and unfogging to ensure that the CLC is at the retina and refined using the standard JCC technique. It is important to note that the above procedure is only effective in a noncyclopeged or nonpresbyopic patient because the active use of accommodation is necessary in practice to found the principal meridians of the eye.

Although the graphical analysis has been carried out in the plane MC_{0} for this particular example, for an arbitrary refractive error whose principal meridians are not 90/180° or 45/135°, the study can not be restricted to a plane, and it is necessary to consider the whole three-dimensional space.

As we have just seen, the classical stenopaic slit refraction technique is based on the detection of one of the principal meridians of the eye, the one that coincides with the orientation of the slit providing the best visual acuity. The other principal meridian can not be clinically determined with the classical technique. ^{18} However, because the stenopaic slit refraction deals with high astigmatic refractive errors, the principal meridians must be properly detected and bracketed. In this way, it could be interesting to design a procedure to clinically locate both orthogonal principal meridians. We propose two alternative procedures. In the first one, the refractive error is converted into a myopic astigmatism to determine the meridians providing the best and the worst visual acuity. It begins by detecting the best vision meridian (β_{1}) in the same way as the traditional procedure states (suppose that L is the lens that brings the CLC to the retina). Then, the eye is fogged and unfogged with spheres until the best acuity is reached by adding a spherical lens of power S over the lens L (L + S). Because the CLC was at the retina, the value of the astigmatism is 2S. Therefore, with the lens S, the focal line that previously was behind the retina is now on it, and the other one is in front of it with a relative dioptric power −2S. Next, the slit is rotated until the meridian that gives the worst acuity (β_{2}) is reached (of course, if β_{1} was well determined, β_{2} must be 90° away from it). Then, the final correction is +S − 2S × β_{1}. The other possible procedure converts the refractive astigmatic error into a hyperopic astigmatism by placing before the eye a lens of power L − S. The difference between this procedure and the previous one is that now the principal meridians are detected by determining the two slit positions that provide better vision.

##### BARNES SUBJECTIVE REFRACTION METHOD

A novel subjective procedure that uses the principle of astigmatic decomposition was proposed by Barnes. ^{19} The first step of this technique consists of the determination of the equivalent sphere correction by placing the CLC at the retina. Because this situation will be maintained until the end of the procedure by continuously adjusting the spherical power, the analysis in the three-dimensional space can be restricted to the JCC plane (M = 0 in Fig. 2). In other words, in the analysis of this clinical procedure, we can ignore the spherical equivalent power of the spherocylindrical refractive errors and consider only the C_{0} and C_{45} components. The fixation test in the Barnes method typically consists of three crossed lines separated by 3.5°. At the beginning, the central line of the test is positioned in either the 45° or 135° meridian. Then, negative cylinder power is introduced along either the 90° or 180° meridian until the central line of the target appears as clear as possible. Without removing the cylinder previously introduced, the test is then rotated until the central line is vertical. The spherical power is readjusted in such a way that the lines remain as clear as possible. The next step consists of introducing negative cylinder power in the 45° or 135° meridian, depending on which one makes the lines clearest. At the end, the final sphere is refined to reach the best acuity. The rationale of this procedure, in principle abstract, can be easily analyzed in the framework of the power vector space. The whole procedure profits the orthogonality of the three components of the power spherocylindrical vector. Suppose that the refractive error is C_{E} × α. In the first step, a cylinder is placed at 90° or 180° until the target, oriented along 45° or 135°, appears as clear as possible. As can be seen in Fig. 8, with a cylinder C_{180}, the correcting lens C_{E} is partially achieved, leaving a residual refractive error C_{R} = C_{E} − C_{180} along the C_{45} axis. Because the test chart is positioned at 45°, the best vision is achieved when C_{R} results in a vector along this axis. As mentioned, the best vision means the smallest resultant vector length C_{R}: any other combination of C_{E} and −C_{180} (or C_{E} and −C_{90}) would give a longer C_{R} vector. Without removing C_{180}, the next step consists of introducing cylindrical power in the 45° or 135° meridian (C_{45} in our case). This cylinder combined with C_{180} gives the correcting cylinder C_{E}, i.e., C_{E} − (C_{45} + C_{180}) = 0. Therefore, using the Barnes method, the correcting lens is achieved by obtaining the spherical equivalent M and the values of both C_{0} and C_{45} components. The standard notation can be obtained by use of Equation 8.

Equation 8 Image Tools |
Figure 8 Image Tools |

As an alternative procedure, we propose to perform the Barnes method without any specific fixation test but with a conventional visual acuity test chart, using spherical lenses, and with the aid of a stenopaic slit. In fact, because the meridians of 45° to 135° and 90° to 180° can be isolated with the slit, the components C_{0} and C_{45} of the correcting cylinder can be found independently by performing refraction only along these two meridians.

##### JACKSON CROSS-CYLINDER PROCEDURE

For the sake of completeness, the JCC technique is also revisited here in the framework of the three-dimensional power space. The use of vector representation of power clarifies the steps used in the refinement of both axis and power of an astigmatic correction. Bearing in mind that the JCC clinical procedure is performed by maintaining the CLC at the retina, the analysis of its working principles in the power space is restricted to the JCC plane. This fact will naturally lead us to the two-dimensional vector addition method used by Keating. ^{20}

As in clinical practice, let us first consider the procedure to refine the cylinder axis. The cylinder axis is determined by placing the ± axes of the JCC lens at an angle of 45° with the correcting cylinder. Then, the JCC is flipped and the patient is required to compare the images at both JCC positions. If we are looking for a minus cylinder refraction, the trial cylinder lens and also the JCC axis are rotated toward the minus JCC axis until the position that provide best vision. This procedure is repeated until no difference between the images can be detected. This equality situation can be considered as the reference point to understand the axis refinement procedure. Suppose that the astigmatic refractive error is (C_{E} × α) and the trial cylinder lens in front of the eye is (C_{C} × β). Then, we assume that the cylinder to be detected with the JCC is (C_{C} × β) − (C_{E} × α) = (C_{C} × β) + (−C_{E} × α). The representation of this situation in the JCC plane of the dioptric power space is given in Fig. 9. In the same figure is also represented the action of the JCC in the two flip positions, i.e., when the minus axis of the JCC is at +45° to the trial cylinder lens, position 1, and when it is at −45°, position 2. The resultant cylinder for each position of the JCC is then the addition of the three vectors: −C_{E}, C_{C}, JCC-1, represented as R_{1} in Fig. 9; and −C_{E}, C_{C}, JCC-2, depicted as R_{2} in the same figure. If α = β, that is to say, if the trial cylinder is properly located, then the action of the JCC, as illustrated in Fig. 9, will produce a resultant cylinder of the same magnitude (R_{1} = R_{2}) for the two flip positions of the JCC. In this case, the patient will show no preference for one flip position over the other because the foci of the resultant astigmatism in each case are equidistant on each side of the retina but obliquely positioned relative to the eye’s original ones. On the other hand, if the trial cylinder axis is not accurately located (α ≠ β), then, as depicted in Fig. 10, the modulus of the resultant cylinder R_{1} is lower than R_{2}. The patient reports better vision when the negative axis of the JCC is in position 1. Of course, to reach the reference situation shown in Fig. 9, the minus trial cylinder (erroneous correcting cylinder) must be rotated toward the minus axis of the JCC that provided the best vision. This step is precisely the one stated by the clinical procedure. The angle of rotation of the trial cylinder depends on its magnitude: as its power increases, the suggested initial rotation diminishes. ^{21} However, for simplicity, the angle of rotation can be fixed, and the final situation (R_{1} and R_{2} vectors of equal length) is reached after several approaching steps. Each step starts with the negative axis of the JCC at 45° from the C_{C} axis such that the rotation of the trial cylinder must be accompanied by a rotation of the same angle of the JCC. In fact, in almost all modern phoropters, the rotation of the JCC is synchronized with the rotation of the trial cylinder.

Figure 9 Image Tools |
Figure 10 Image Tools |

A more detailed description of the JCC procedure in the power space concerning not only the axis but also the power cylinder refinement is available. ^{22}

#### DISCUSSION

Although the vectorial analysis of some refraction methods, such as the Barnes method or the Jackson cross-cylinders, can be done in a single plane, most optometric procedures can not be fully described unless a three-dimensional space of powers is used. This is the case for the stenopaic slit refraction, for which we have shown that the analysis in the three-dimensional space is useful to identify the residual refractive error along the whole optometric procedure. Furthermore, in light of the performed analysis, we have proposed some improvements to the traditional procedures at issue.

It is worth mentioning that the proposed analysis is not restricted to the particular techniques considered here. In fact, it can be used to revise other optometric procedures or even to describe the principle of operation of various optometric instruments.

#### ACKNOWLEDGMENTS

This work was supported by the project GV99-100-1-01 of the Conselleria de Cultura, Educaci’o i Ci‘encia. Generalitat Valenciana, Spain. L. Muñoz-Escrivá gratefully acknowledges financial support from the “Cinc segles” Program, Universitat de Val‘encia, Spain. We thank J. Jonkman for his useful comments on revising the manuscript.

Received April 13, 2000; revision received October 13, 2000.