**In this study we investigated the impact of accommodation on axial and oblique astigmatism along 12 meridians of the central 30° of visual field and explored the compensation of corneal first-surface astigmatism by the remainder of the eye's optical system. Our experimental evidence revealed no systematic effect of accommodation on either axial or oblique astigmatism for two adult populations (myopic and emmetropic eyes). Although a few subjects exhibited systematic changes in axial astigmatism during accommodation, the dioptric value of these changes was much smaller than the amount of accommodation. For most subjects, axial and oblique astigmatism of the whole eye are both less than for the cornea alone, which indicates a compensatory role for internal optics at all accommodative states in both central and peripheral vision. A new method for determining the eye's optical axis based on visual field maps of oblique astigmatism revealed that, on average, the optical axis is 4.8° temporal and 0.39° superior to the foveal line-of-sight in object space, which agrees with previous results obtained by different methodologies and implies that foveal astigmatism includes a small amount of oblique astigmatism (0.06 D on average). Customized optical models of each eye revealed that oblique astigmatism of the corneal first surface is negligible along the pupillary axis for emmetropic and myopic eyes. Individual variation in the eye's optical axis is due in part to misalignment of the corneal and internal components that is consistent with tilting of the crystalline lens relative to the pupillary axis.**

*sectional astigmia*or

*astigmatic accommodation*(Beck, 1965; Brzezinski, 1982; McFadden, 1925) but leaves open the question of which type of astigmatism is responsible. The available evidence suggests that axial astigmatism changes more than oblique astigmatism when the eye accommodates since foveal changes (Cheng et al., 2004; Millodot & Thibault, 1985; Radhakrishnan & Charman, 2007; Ukai & Ichihashi, 1991) are larger than changes in the near peripheral visual field (eccentricity <30°) (Calver, Radhakrishnan, Osuobeni, & O'Leary, 2007; Davies & Mallen, 2009; Lundström, Mira-Agudelo, & Artal, 2009; Mathur, Atchison, & Charman, 2009; Smith, Millodot, & McBrien, 1988; Whatham et al., 2009), although larger changes have been reported in the far periphery (Smith et al., 1988). To help resolve this issue, the present study monitored changes in the axial and oblique components of ocular astigmatism over a range of accommodative states.

*z*axis coinciding with the peripheral LoS and the measurement axis of the aberrometer. From the Zernike coefficients for astigmatism (

*C*

_{2}

^{2},

*C*

_{2}

^{−2}) for the natural pupil size we computed dioptric power vectors

**J**= (

*J*

_{0},

*J*

_{45}) in Cartesian form according to Equation 1 where

*R*is the radius of the natural pupil in mm. The conversion from Cartesian to polar form is given by Equation 2 where

*J*is the magnitude of astigmatism (half of the cylinder power) and

*ϕ*(the meridian of maximum positive power) is the axis of astigmatism. In this report we refer to a collection of power vectors obtained from the 37 aberrometer locations as a visual field map of the astigmatic vector

**J**. Graphical conventions for displaying visual field maps of astigmatism are described in the 1 (Figure A1). Unlike clinical conventions, astigmatism in this report refers to the eye (not the lens used to correct the eye) as recommended by ANSI standard Z80.28 (ANSI, 2010). This explains why Zernike coefficients and power vector values have the same sign in Equation 1. For example, a positive value of

*J*

_{0}in this report indicates “against-the-rule” astigmatism in clinical terminology.

^{−3}D/deg

^{2}. We presume the measurements were obtained for a relaxed eye because the fixation target in both instruments is a poor stimulus for accommodation (flashing red light for Lenstar and black disk for Medmont).

*p*(

*p*= 1 is a circle, with other positive

*p*values being ellipses, and

*p*< 0 are hyperbolas; Rabbetts, 2007) and curvature

*c*(first term in Equation 3). By subtracting the conic surface from the elevation map, the residual shape (second term in Equation 3) was then fit with Zernike polynomials up to the 6th order (Schwiegerling, Greivenkamp, & Miller, 1995) for the corneal diameter (

*D*= 2

*R*, where

_{C}*R*is the measured corneal radius) reported by the topographer. For optical modeling purposes the elevation height map was then expressed in polar coordinates by where

_{C}*z*is the corneal height in mm,

*p*is the unitless conic constant,

*c*is curvature in mm

^{−1},

*C*is

_{i}*i*th Zernike coefficient in mm,

*Z*is

_{i}*i*th Zernike polynomial,

*r*is radial distance from corneal apex (the intersection of measurement axis and cornea) in mm,

*θ*is the meridional angle in radians, and

*ρ*=

*r*/

*R*is the normalized radial coordinate of points on the corneal surface.

_{C}*internal aberrations*refer to those ocular aberrations not accounted for by refraction at the anterior surface of the cornea. According to this definition, refraction by the posterior corneal surface, propagation of aberrated wavefronts through the anterior chamber (Roselló, Thibos, & Micó, 2014), and intrinsic lens aberrations are all factors that contribute to internal aberrations. No attempt was made to separately quantify these contributing factors. Since ocular and corneal aberrations were both specified in the entrance pupil plane for the same pupil diameter and referenced to a common axis (LoS) by the method described in the 1, internal aberrations could be computed for each state of accommodation by a vector subtraction of corneal first-surface aberrations (determined by ray tracing through customized optical models) from whole-eye aberrations (determined by empirical aberrometry) at each point in the visual field. For reporting purposes, computed Zernike coefficients (i.e., RMS wavefront error in microns) for internal astigmatism were converted by Equation 1 to a power vector description in diopters. To further decompose ocular, corneal first-surface, or internal astigmatism into axial and oblique subtypes required an appropriate definition and method for locating the optical axis as described next.

**J**

_{oblique}by performing a power vector subtraction (Liu & Thibos, 2016) of axial astigmatism

**J**

_{axial}from total astigmatism

**J**

_{total}for every field location, using the same vector equation, ocular, corneal and internal astigmatisms were computed from Zernike aberration coefficients and then decomposed into axial and oblique component for evaluation.

*J*(diopters) across meridians for fitting with the quadratic Equation 5 where

*ε*is the radial eccentricity (degrees from the optical axis) and σ is regression coefficient (Diopters/deg

^{2}).

*ocular*(i.e., the whole eye, as measured by aberrometry),

*corneal*(as measured by videokeratography), and

*internal astigmatism*(computed by subtracting corneal from total) representing the remainder of ocular astigmatism not accounted for by refraction at the anterior corneal surface. Within each of these three categories, we decompose the

*total*astigmatism into

*axial*and

*oblique*components using Methods Equation 4. This organized collection of results is described first for a typical subject (RH) to demonstrate how the sequence of optical analysis proceeds for an individual eye. We then present population results to show trends and individual variability within and between populations of emmetropic and myopic eyes.

*SD*= 0.37° horizontally; mean = 1.8°,

*SD*= 0.92° vertically). By definition, oblique astigmatism is zero at the optical axis and therefore the interpolated astigmatism at the mean optical axis location is our best estimate of axial astigmatism.

*J*

_{0}mean = −0.18,

*SD*= 0.03 D;

*J*

_{45}mean = −0.12,

*SD*= 0.03 D) and clustered together with high concentration factor (|B| = 0.99) about the mean (indicated by the red arrow) in the scatter map of Figure 2. For this subject the mean magnitude of axial astigmatism at the optical axis was 0.22 D axis 107° with negligible fluctuation over the 6 D range of accommodative demand. Inspecting the trace of axial astigmatism as this subject accommodated showed no systematic changes. Assuming accommodation has no effect on corneal astigmatism, we infer from this result that differences in restraining forces on the crystalline lens resulting from ciliary muscle action were symmetrically exerted without changing surface toricity, tilt, or displacement in this eye.

*σ*for the averaged oblique astigmatism = 1.07 × 10

^{−3}D/deg

^{2}, which corresponds to approximately

*J*= 0.24 D oblique astigmatism at 15° eccentricity.

*J*

_{0}< 0). Comparing the relaxed state (accommodation demand 0 D) with the most accommodated state (accommodation demand 6 D), reveals that both populations tended to shift slightly towards against-the-rule astigmatism when focusing on near targets. However, the lack of a systematic trend between the limits of 0 and 6 D accommodation suggests that neither population exhibited a definitive variation as the eye accommodated. To verify this observation, the power vector of accommodative axial astigmatism obtained by subtracting axial astigmatism of the relaxed state from the axial astigmatism of the most accommodated state is shown in the lower panel of Figure 4. Most noticeable from this graph is the relatively small difference in population means (filled symbols) compared to the larger intersubject variability in both populations. In addition, the coordinate origin (which corresponding to relaxed state) falls within the 95% confidence ellipses (dashed lines) of mean accommodative astigmatism, suggesting that the null hypothesis that accommodation has no effect on axial astigmatism cannot be rejected (Thibos et al., 1997).

*σ*that quantifies the rate of change of oblique astigmatism with visual field eccentricity relative to the optical axis. The population mean of this regression coefficient declined slightly with accommodation, more for emmetropic eyes than for myopic eyes as plotted in Figure 6. Emmetropic eyes showed more individual variability than myopic eyes. At 15° eccentricity, the difference in oblique astigmatism between the most relaxed and most accommodative state was 0.04 D for emmetropic eyes and 0.01 D for myopic eyes. Although the emmetropic population exhibited more oblique astigmatism than myopic eyes at all accommodative states, the maximum difference between populations was only 0.04 D at 15° eccentricity. Along the LoS (approximately 5° from the optical axis), about 0.06 D oblique astigmatism was present for both groups.

*σ*= 1.50 × 10

^{−3}D/deg

^{2}) was greater than that of the whole eye (1.07 × 10

^{−3}D/deg

^{2}), indicative of compensation by internal optics. This compensation is rather complicated to compute, however, since the local optical axis of the cornea (cross symbol) deviated from optical axis of the whole eye (diamond symbol) about 2.8° for subject RH. This implies that the optical axes of cornea and lens are not collinear for this subject, which was true also for other subjects.

*σ*= 0.66 × 10

^{−3}D/deg

^{2}) reduced corneal oblique astigmatism (

*σ*= 1.50 × 10

^{−3}D/deg

^{2}) roughly by one third, resulting in an intermediate level of ocular oblique astigmatism (

*σ*= 1.07 × 10

^{−3}D/deg

^{2}). However, unlike total astigmatism, the quadratic coefficients of oblique astigmatism cannot be added linearly in principle because they are referenced to local optical axes that are not aligned with each other. For RH, the local optical axis of the lens (triangle symbol) is superior to the corneal axis (cross symbol), and the angle between them is 6.86°. This misalignment explains the deviation of the ocular optical axis from either of the local optical axes.

*J*

_{0}< 0 for the cornea but

*J*

_{0}> 0 internally. This change in sign makes it clear why their sum (i.e., ocular axial astigmatism) is less than the corneal contribution, which we summarize by terms such as “compensation” or “balancing.” In general, the axes of corneal and internal astigmatism are not necessarily orthogonal, so a vector sum must be computed to assess the effect of their interaction on the magnitude and axis of ocular astigmatism for other eyes in the study population.

*J*

_{0}components have opposite signs, their

*J*

_{45}components have opposite signs, and their astigmatic axes are orthogonal. These results confirm that subject RH is typical of the other emmetropic subjects in our study population, as well as the myopic population as shown in Figure 10. Thus, we are led to conclude that at least partial balancing of corneal axial astigmatism by internal axial astigmatism is commonplace, in the spirit of Javal's Rule of clinical optometry and ophthalmology, for all states of accommodation.

*p*= 0.67,

*t*test), which justifies pooling across populations to show frequency histogram of COC for all 33 subjects in Figure 11. Most eyes showed significant compensation, with only two of 33 eyes exhibiting reinforcement. From these data we conclude that oblique astigmatism of the hypothetical average adult eye is 18% less than that of the cornea because of compensation by internal optics.

**B**| = 0.997 for myopic eyes and |

**B**| = 0.999 for emmetropic eyes), and consequently the nonzero values of the mean angle α′ are statistically significant for both populations (Rayleigh test). Thus, for the purposes of defining a hypothetical average eye that represents the population mean, the present results confirm previous reports that the optical axis is about 5° away from LoS in the temporal visual field (Atchison, Smith, & Smith, 2000; Shen, Clark, Soni, & Thibos, 2010), which suggests methodological differences in defining and locating the optical axis in previous studies and in ours are immaterial.

*λ*,

_{x}*λ*) was calculated from corneal topography data using trigonometry (see the 1). Since both axes are in reference to a common LoS, their mutual alignment for an individual eye is obtained by subtraction (

_{y}*λ*,

_{x}*λ*), which is shown graphically in Figure 14 (top panel) as a point in a two-dimensional scatter plot. The population mean of these directional data for the emmetropic eyes and also for the myopic eyes are both within 1° of the origin. Moreover, the 95% confidence ellipses for these mean directions includes the coordinate origin, which confirms that the corneal optical axis and the pupillary axis are not significantly different (Hotelling's

_{y}*T*

^{2}test

*p*= 0.21 for myopic population, and

*p*= 0.22 for emmetropic population). These results confirm that the pupillary axis derived from customized eye models is essentially identical to the corneal optical axis of zero oblique astigmatism. The same conclusion was reached by the two-sample version of Hotelling's

*T*

^{2}test (

*p*= 0.56) which confirms that oblique astigmatism of the corneal first surface is zero along the pupillary axis.

*T*

^{2}test

*p*= 0.32), so further analysis was performed on the combined populations. Mean α′ for all subjects was 2.1° temporal and 1.2° inferior with respect to

^{−3}D/deg

^{2}, standard deviation = 0.23 × 10

^{−3}D/deg

^{2}). Extrapolation of our results indicated a greater degree of oblique astigmatism may be expected in the far periphery than has been reported previously.

*SD*= 1.1 mm for relaxed eyes; mean = 6.4 mm,

*SD*= 1.2 mm for accommodating eyes). The average pupil constriction for the myopic population was 1.4 mm (relaxed mean = 6.9 mm, SD = 0.9; accommodating mean = 5.6 mm, SD = 1.1). These levels of pupil constriction are small compared to the constriction and concomitant decentration produced by the corneal topographer, which we argue above had negligible effect on astigmatism. We conclude, therefore, that small variations in pupil size produced by accommodation in our study would be expected to have a minor impact on nonconcentric pupil constriction and negligible impact on corneal astigmatism.

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*z*axis that is perpendicular to a tangent plane for specifying test locations in Cartesian (

*x*,

*y*) or in polar (eccentricity,

*ε*and meridian,

*θ*) coordinates. The line joining a test point with the center of the left eye's entrance pupil can be regarded as a peripheral LoS that coincides with the corresponding measurement axis (MA) of the scanning aberrometer. For graphical purposes in this report, visual field maps are oriented and labeled as if seen by the subject's left eye (and also the reader's left eye), with the top of the map occupying the superior field and the left side occupying the temporal field. Mathematical coordinates of the field maps are specified in Cartesian (

*x*,

*y*) form with positive

*x*values indicating nasal visual field and positive

*y*values indicating inferior field. The same maps can also be interpreted in retinal coordinates, in which case positive

*x*values indicate temporal retina and positive

*y*values indicate superior retina for the location of the image of the test spot, which acts as a retinal beacon reflecting light out of the eye for measurement by the aberrometer.

*O*, which is connected to test point

*P*by a vector (blue dashed line). A candidate location for the optical axis is shown by the red circle labeled

*O*′, which is connected to the test point

*P*by the red dashed line. For each test location, the astigmatism vector is projected onto the corresponding red line and the scalar sum of these projected lengths for all field locations is computed. Using an iterative algorithm, the candidate location

*O*′ moves around the visual field until this sum of projections is maximized. In principle, this maximum location is

*O*, the optical axis, because the axis of oblique astigmatism is always radial and therefore produces the largest possible radial projection for every field location. The method works best when axial astigmatism is weak or absent, so the maps were preconditioned by subtracting the average across all states of accommodation of the space average value of astigmatism across visual field.

*θ*, respectively. FV = 59.7 mm. ED was estimated with the deviation of entrance pupil center (E) from center of Placido rings (D) in videokeratoscopic images. The entrance pupil depth VD could be computed from measured anterior chamber depth VL by considering refraction by the cornea and assuming refractive index 1.3375, where

*c*is the mean power of anterior cornea reported by topographer.

_{m}*z*axis was aligned with the LoS, the anterior corneal surface sag was represented by Equation 3, tilted and decentered appropriately, the stop aperture (physical pupil) was placed at the position determined by anterior chamber depth measurements. The entrance pupil was set equal to the minimum pupil size during ocular aberration measurements, and the object was placed at infinity. The posterior cornea was ignored and refractive index of image space was set equal to 1.3700 as predicted by chromatic dispersion (Navarro, Santamaría, & Bescós, 1985) of the cornea at 850 nm. The model's imaging surface (i.e., model retina) was a sphere with 12 mm radius of curvature positioned to eliminate paraxial focusing errors along the foveal LoS. Then the numerical ray tracing was performed along the same scanning pattern carried out in our ocular aberration experiment. The resultant Zernike coefficients for corneal aberrations were scaled appropriately to match the eye's pupil size during ocular aberration measurements (which varied with accommodation), and the results expressed as power vectors for corneal astigmatism as described previously (Liu & Thibos, 2016).

*ε*,

*θ*) but then are transformed into directional cosines for computing statistics (see chapter 8 of Mardia, 2014). The resulting mean direction was then projected back onto the tangent plane for display.

**J**with a magnitude and direction equal to twice the axis angle. To compute the mean of

*N*power vectors we divide their vector sum by

*N,*

**J̄**=

**B**=