1040-5488/14/9105-0582/0 VOL. 91, NO. 5, PP. 582Y587 OPTOMETRY AND VISION SCIENCE Copyright * 2014 American Academy of Optometry
ORIGINAL ARTICLE
Visual Field Coordinates of Pupillary Circular Axis and Optical Axis David A. Atchison*, Ankit Mathur†, Marwan Suheimat‡, and W. Neil Charman§
ABSTRACT Purpose. We term the visual field position from which the pupil appears most nearly circular as the pupillary circular axis (PCAx). The aim was to determine and compare the horizontal and vertical coordinates of the PCAx and optical axis from pupil shape and refraction information for only the horizontal meridian of the visual field. Methods. The PCAx was determined from the changes with visual field angle in the ellipticity and orientation of pupil images out to T90 degrees from fixation along the horizontal meridian for the right eyes of 30 people. This axis was compared with the optical axis determined from the changes in the astigmatic components of the refractions for field angles out to T35 degrees in the same meridian. Results. The mean estimated horizontal and vertical field coordinates of the PCAx were j5.3 (T1.9) and j3.2 (T1.5) degrees compared with j4.8 (T5.1) and j1.5 (T3.4) degrees for the optical axis, respectively. The vertical coordinates of the two axes were just significantly different (p = 0.03), but there was no significant correlation between them. Only the horizontal coordinate of the PCAx was significantly related to the refraction in the group. Conclusions. On average, the PCAx is displaced from the line-of-sight by about the same angle as the optical axis, but there is more intersubject variation in the position of the optical axis. When modeling the optical performance of the eye, it appears reasonable to assume that the pupil is circular when viewed along the line-of-sight. (Optom Vis Sci 2014;91:582Y587) Key Words: astigmatism, asymmetry, optical axis, peripheral vision, pupillary circular axis, pupil shape, visual field
I
t is usually assumed that, in peripheral vision, the eye’s entrance pupil approximates to an ellipse, with the minor axis oriented along the meridian of the visual field from which it is observed. Measurements by many authors along the horizontal field meridian suggest that the value (A) given by dividing the length of the minor axis of the ellipse by that of the major axis as a function of the visual field angle E varies somewhat more slowly than cos E. Moreover, along this meridian, the maximum of A is displaced slightly from zero, typically to a position at a few degrees into the temporal field. This is attributed to a lack of optical symmetry of the eye about its line-of-sight.1 The question arises: does this asymmetry occur only in the horizontal meridian or does
*PhD, DSc, FAAO † PhD, FAAO ‡ PhD § DSc Institute of Health & Biomedical Innovation and School of Optometry & Vision Science, Queensland University of Technology, Kelvin Grove, Queensland, Australia (DAA, AM, MS); and Faculty of Life Sciences, University of Manchester, Manchester, United Kingdom (WNC).
it also occur in the vertical meridian? We show here that any vertical displacement can be deduced from measurements made along the horizontal field meridian. The key point is that, if there is no vertical displacement, the minor axes of the pupil ellipses will always remain horizontal along the horizontal field meridian and the major axes vertical, but any vertical asymmetry will result in a slight tilt in the axes of the ellipses. To explore this possibility, suitable data are available from a recent study by Mathur et al.1 in which pupil shape was determined in 30 eyes as viewed along the horizontal visual field. Pupil shape at field angle E was described as A, the pupil diameter in the ‘‘horizontal’’ meridian divided by that in the ‘‘vertical,’’ where ‘‘horizontal’’ refers to the principal meridian of the best-fitting ellipse within 45 degrees of the horizontal meridian. Cosine fits were made to the A values as:
AðEÞ ¼ D cos½ðEjAÞ=E
ð1Þ
where D is the amplitude of the fit, E is the visual field angle (negative/positive for temporal/nasal visual field), A is the peak of the fit relative to the center of the visual field, and 180E is half the period of the fit. All angles are in degrees.
Optometry and Vision Science, Vol. 91, No. 5, May 2014
Copyright © American Academy of Optometry. Unauthorized reproduction of this article is prohibited.
Visual Field Coordinates of Pupillary Circular Axis and Optical AxisVAtchison et al.
Two components of pupil ellipticity were determined, B is the horizontal-vertical component and C is the oblique component, as:
B ¼ ð1jAÞcos½2ð5 j 90Þ
ð2Þ
C ¼ ð1jAÞsin½2ð5 j 90Þ
ð3Þ
where 5 is the orientation of the major axis of the pupil ellipse relative to the horizontal field axis, as viewed by an observer for a counterclockwise rotation from the right side of the pupil . An idealized example of the variation of A, B, and C across the field for particular values of parameters D, E, A, and E is shown in Fig. 1A, whereas Fig. 1B shows an example of the pupil ellipticity and orientation 5 at a particular field angle. Note that, if the minor axis of the ellipse was always horizontal (5 = 90 degrees), we would have B = (1 j A) and C = 0 at all positions along the horizontal meridian. Combining all subjects’ data, the study gave the fit to A as:
AðEÞ ¼ 0:992 cos½ðE þ 5:3Þ=1:121
ð4Þ
This represents a cosine function with an amplitude close to unity, a peak (j)5.3 degrees into the temporal visual field, and a width greater by 12% than that predicted by the cosine of the visual field angle. Combining all subjects’ data, a linear fit to C as:
CðEÞ ¼ þ0:00072E j 0:0120
ð5Þ
was obtained. The authors pointed out that the non-zero, positive slope indicates that there is a vertical coordinate to the visual field position where the pupil appears circular and that this position (Ex0, Ey0) is below the fixation axis (or line-of-sight) that has field coordinates (0, 0), although they did not determine the magnitude of this offset. We term the visual field position at which the pupil
583
appears most nearly circular as the pupillary circular axis (PCAx) to distinguish it from the pupillary axis, which is the axis of alignment of the pupil center with the corneal center of curvature. We now show a procedure for estimating the vertical coordinate of the PCAx from this type of pupil shape data along the horizontal visual field. We have used a broadly similar procedure previously to determine the vertical coordinate of the optical axis from refractions along the horizontal visual field.2 The present study has clinical relevance because it is usually assumed that the pupil is close to circular along the line-of-sight and a failure for this to be the case will affect retinal image quality.
METHODS Pupillary Circular Axis Charman and Atchison2 developed equations for determining the visual field positions about which aberrations were symmetric when aberration data were known for the horizontal and vertical meridians of the visual field or for only one of these meridians. It was assumed that this position corresponded to the optical axis. In the present case, the problem is that the horizontal and vertical visual field angles (Ex, Ey) are referenced to the line-of-sight coordinates (0, 0) rather than to the PCAx at (Ex0, Ey0) (Fig. 2). To allow for this, the generalized pupil shape equation should be:
2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 ðEx jEx0 Þ2 þ ðEy jEy0 Þ2 5 PðEx ; Ey Þ ¼ D cos4 E
ð6Þ
FIGURE 1. (A) Idealized pupil diameter ratio A, horizontal/vertical component of pupil ellipticity B, and oblique component of pupil ellipticity C in one case. The fit for E þ 10. (B) At (+)70 degrees in the nasal visual field, the major axis of the pupil ellipse is at 97.1 degrees to the horizontal equation 1 is AðEÞ ¼ 0:987 cos 1:10 meridian. A color version of this figure is available online at www.optvissci.com. Optometry and Vision Science, Vol. 91, No. 5, May 2014
Copyright © American Academy of Optometry. Unauthorized reproduction of this article is prohibited.
584 Visual Field Coordinates of Pupillary Circular Axis and Optical AxisVAtchison et al.
where P (Ex, Ey) is the two-dimensional equivalent of equation 1 for visual field position (Ex, Ey). A regular, or horizontal-vertical, component to the pupil ellipticity is given by:
RðEx ; Ey Þ ¼ ½1jPðEx ; Ey Þcos½2ð5 j 90Þ
ð7Þ
Replacing the right-hand expressions of equations 11 and 12 for the left-hand expressions in equations 7 and 8 gives:
RðEx ; Ey Þ ¼ ½1jPðEx ; Ey Þ
ðEx jEx0 Þ2 j ðEy jEy0 Þ2 ðEx jEx0 Þ2 þ ðEy jEy0 Þ2
ð13Þ
and an oblique component to the pupil ellipticity is given by:
OðEx ; Ey Þ ¼ ½1jPðEx ; Ey Þsin½2ð5 j 90Þ
ð8Þ
Equations 7 and 8 are the two-dimensional equivalents to equations 2 and 3. Using Fig. 2, we can derive expressions for cos [2(5 Y 90)] and sin[2(5 Y 90)] in terms of visual field positions for use in equations 7 and 8, respectively. It can be seen that for any visual field point (Ex, Ey): 5
¼ 90 þ F
where
F ¼ tanj1
Ey jEy0 Ex jEx0
ð9Þ
ð10Þ
ðEx jEx0 Þ2 þ ðEy jEy0 Þ2
ðEx jEx0 ÞðEy jEy0 Þ ðEx jEx0 Þ2 þ ðEy jEy0 Þ2
ð14Þ
Substituting the right-hand side of equation 6 for P (Ex, Ey) in equations 13 and 14, we have: 8 2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi39 < ðEx jEx0 Þ2 þ ðEy jEy0 Þ2 = 4 5 RðEx ; Ey Þ ¼ 1jD cos ; : E
ðEx jEx0 Þ2 þ ðEy Ey0 Þ2
ð15Þ
2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi39 ðEx jEx0 Þ2 þ ðEy jEy0 Þ2 = 5 OðEx ; Ey Þ ¼ 2 1jD cos4 ; : E ð11Þ ðEx jEx0 ÞðEy jEy0 Þ ðEx jEx0 Þ2 þ ðEy jEy0 Þ2
sin½2ð5 j 90Þ ¼ sinð2FÞ ¼ 2sinFcosF ¼2
ðEx jEx0 Þ2 þ ðEy jEy0 Þ2
8
ORIGINAL ARTICLE
Visual Field Coordinates of Pupillary Circular Axis and Optical Axis David A. Atchison*, Ankit Mathur†, Marwan Suheimat‡, and W. Neil Charman§
ABSTRACT Purpose. We term the visual field position from which the pupil appears most nearly circular as the pupillary circular axis (PCAx). The aim was to determine and compare the horizontal and vertical coordinates of the PCAx and optical axis from pupil shape and refraction information for only the horizontal meridian of the visual field. Methods. The PCAx was determined from the changes with visual field angle in the ellipticity and orientation of pupil images out to T90 degrees from fixation along the horizontal meridian for the right eyes of 30 people. This axis was compared with the optical axis determined from the changes in the astigmatic components of the refractions for field angles out to T35 degrees in the same meridian. Results. The mean estimated horizontal and vertical field coordinates of the PCAx were j5.3 (T1.9) and j3.2 (T1.5) degrees compared with j4.8 (T5.1) and j1.5 (T3.4) degrees for the optical axis, respectively. The vertical coordinates of the two axes were just significantly different (p = 0.03), but there was no significant correlation between them. Only the horizontal coordinate of the PCAx was significantly related to the refraction in the group. Conclusions. On average, the PCAx is displaced from the line-of-sight by about the same angle as the optical axis, but there is more intersubject variation in the position of the optical axis. When modeling the optical performance of the eye, it appears reasonable to assume that the pupil is circular when viewed along the line-of-sight. (Optom Vis Sci 2014;91:582Y587) Key Words: astigmatism, asymmetry, optical axis, peripheral vision, pupillary circular axis, pupil shape, visual field
I
t is usually assumed that, in peripheral vision, the eye’s entrance pupil approximates to an ellipse, with the minor axis oriented along the meridian of the visual field from which it is observed. Measurements by many authors along the horizontal field meridian suggest that the value (A) given by dividing the length of the minor axis of the ellipse by that of the major axis as a function of the visual field angle E varies somewhat more slowly than cos E. Moreover, along this meridian, the maximum of A is displaced slightly from zero, typically to a position at a few degrees into the temporal field. This is attributed to a lack of optical symmetry of the eye about its line-of-sight.1 The question arises: does this asymmetry occur only in the horizontal meridian or does
*PhD, DSc, FAAO † PhD, FAAO ‡ PhD § DSc Institute of Health & Biomedical Innovation and School of Optometry & Vision Science, Queensland University of Technology, Kelvin Grove, Queensland, Australia (DAA, AM, MS); and Faculty of Life Sciences, University of Manchester, Manchester, United Kingdom (WNC).
it also occur in the vertical meridian? We show here that any vertical displacement can be deduced from measurements made along the horizontal field meridian. The key point is that, if there is no vertical displacement, the minor axes of the pupil ellipses will always remain horizontal along the horizontal field meridian and the major axes vertical, but any vertical asymmetry will result in a slight tilt in the axes of the ellipses. To explore this possibility, suitable data are available from a recent study by Mathur et al.1 in which pupil shape was determined in 30 eyes as viewed along the horizontal visual field. Pupil shape at field angle E was described as A, the pupil diameter in the ‘‘horizontal’’ meridian divided by that in the ‘‘vertical,’’ where ‘‘horizontal’’ refers to the principal meridian of the best-fitting ellipse within 45 degrees of the horizontal meridian. Cosine fits were made to the A values as:
AðEÞ ¼ D cos½ðEjAÞ=E
ð1Þ
where D is the amplitude of the fit, E is the visual field angle (negative/positive for temporal/nasal visual field), A is the peak of the fit relative to the center of the visual field, and 180E is half the period of the fit. All angles are in degrees.
Optometry and Vision Science, Vol. 91, No. 5, May 2014
Copyright © American Academy of Optometry. Unauthorized reproduction of this article is prohibited.
Visual Field Coordinates of Pupillary Circular Axis and Optical AxisVAtchison et al.
Two components of pupil ellipticity were determined, B is the horizontal-vertical component and C is the oblique component, as:
B ¼ ð1jAÞcos½2ð5 j 90Þ
ð2Þ
C ¼ ð1jAÞsin½2ð5 j 90Þ
ð3Þ
where 5 is the orientation of the major axis of the pupil ellipse relative to the horizontal field axis, as viewed by an observer for a counterclockwise rotation from the right side of the pupil . An idealized example of the variation of A, B, and C across the field for particular values of parameters D, E, A, and E is shown in Fig. 1A, whereas Fig. 1B shows an example of the pupil ellipticity and orientation 5 at a particular field angle. Note that, if the minor axis of the ellipse was always horizontal (5 = 90 degrees), we would have B = (1 j A) and C = 0 at all positions along the horizontal meridian. Combining all subjects’ data, the study gave the fit to A as:
AðEÞ ¼ 0:992 cos½ðE þ 5:3Þ=1:121
ð4Þ
This represents a cosine function with an amplitude close to unity, a peak (j)5.3 degrees into the temporal visual field, and a width greater by 12% than that predicted by the cosine of the visual field angle. Combining all subjects’ data, a linear fit to C as:
CðEÞ ¼ þ0:00072E j 0:0120
ð5Þ
was obtained. The authors pointed out that the non-zero, positive slope indicates that there is a vertical coordinate to the visual field position where the pupil appears circular and that this position (Ex0, Ey0) is below the fixation axis (or line-of-sight) that has field coordinates (0, 0), although they did not determine the magnitude of this offset. We term the visual field position at which the pupil
583
appears most nearly circular as the pupillary circular axis (PCAx) to distinguish it from the pupillary axis, which is the axis of alignment of the pupil center with the corneal center of curvature. We now show a procedure for estimating the vertical coordinate of the PCAx from this type of pupil shape data along the horizontal visual field. We have used a broadly similar procedure previously to determine the vertical coordinate of the optical axis from refractions along the horizontal visual field.2 The present study has clinical relevance because it is usually assumed that the pupil is close to circular along the line-of-sight and a failure for this to be the case will affect retinal image quality.
METHODS Pupillary Circular Axis Charman and Atchison2 developed equations for determining the visual field positions about which aberrations were symmetric when aberration data were known for the horizontal and vertical meridians of the visual field or for only one of these meridians. It was assumed that this position corresponded to the optical axis. In the present case, the problem is that the horizontal and vertical visual field angles (Ex, Ey) are referenced to the line-of-sight coordinates (0, 0) rather than to the PCAx at (Ex0, Ey0) (Fig. 2). To allow for this, the generalized pupil shape equation should be:
2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 ðEx jEx0 Þ2 þ ðEy jEy0 Þ2 5 PðEx ; Ey Þ ¼ D cos4 E
ð6Þ
FIGURE 1. (A) Idealized pupil diameter ratio A, horizontal/vertical component of pupil ellipticity B, and oblique component of pupil ellipticity C in one case. The fit for E þ 10. (B) At (+)70 degrees in the nasal visual field, the major axis of the pupil ellipse is at 97.1 degrees to the horizontal equation 1 is AðEÞ ¼ 0:987 cos 1:10 meridian. A color version of this figure is available online at www.optvissci.com. Optometry and Vision Science, Vol. 91, No. 5, May 2014
Copyright © American Academy of Optometry. Unauthorized reproduction of this article is prohibited.
584 Visual Field Coordinates of Pupillary Circular Axis and Optical AxisVAtchison et al.
where P (Ex, Ey) is the two-dimensional equivalent of equation 1 for visual field position (Ex, Ey). A regular, or horizontal-vertical, component to the pupil ellipticity is given by:
RðEx ; Ey Þ ¼ ½1jPðEx ; Ey Þcos½2ð5 j 90Þ
ð7Þ
Replacing the right-hand expressions of equations 11 and 12 for the left-hand expressions in equations 7 and 8 gives:
RðEx ; Ey Þ ¼ ½1jPðEx ; Ey Þ
ðEx jEx0 Þ2 j ðEy jEy0 Þ2 ðEx jEx0 Þ2 þ ðEy jEy0 Þ2
ð13Þ
and an oblique component to the pupil ellipticity is given by:
OðEx ; Ey Þ ¼ ½1jPðEx ; Ey Þsin½2ð5 j 90Þ
ð8Þ
Equations 7 and 8 are the two-dimensional equivalents to equations 2 and 3. Using Fig. 2, we can derive expressions for cos [2(5 Y 90)] and sin[2(5 Y 90)] in terms of visual field positions for use in equations 7 and 8, respectively. It can be seen that for any visual field point (Ex, Ey): 5
¼ 90 þ F
where
F ¼ tanj1
Ey jEy0 Ex jEx0
ð9Þ
ð10Þ
ðEx jEx0 Þ2 þ ðEy jEy0 Þ2
ðEx jEx0 ÞðEy jEy0 Þ ðEx jEx0 Þ2 þ ðEy jEy0 Þ2
ð14Þ
Substituting the right-hand side of equation 6 for P (Ex, Ey) in equations 13 and 14, we have: 8 2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi39 < ðEx jEx0 Þ2 þ ðEy jEy0 Þ2 = 4 5 RðEx ; Ey Þ ¼ 1jD cos ; : E
ðEx jEx0 Þ2 þ ðEy Ey0 Þ2
ð15Þ
2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi39 ðEx jEx0 Þ2 þ ðEy jEy0 Þ2 = 5 OðEx ; Ey Þ ¼ 2 1jD cos4 ; : E ð11Þ ðEx jEx0 ÞðEy jEy0 Þ ðEx jEx0 Þ2 þ ðEy jEy0 Þ2
sin½2ð5 j 90Þ ¼ sinð2FÞ ¼ 2sinFcosF ¼2
ðEx jEx0 Þ2 þ ðEy jEy0 Þ2
8
