**Purpose**:
To provide a tool for calculating radial and tangential retinal magnifications as functions of field angle and retinal shape and to articulate patterns of magnification across the retina for monocular and binocular combinations of prolate-, oblate-, and spherical-shaped retinas.

**Methods**:
Formulae were derived to calculate radial and tangential retinal magnifications (mm/deg) from field angle (degrees), retinal asphericity (unitless conic constant), retinal vertex radius of curvature (mm), and nodal point position (mm). Monocular retinal magnifications were determined for eyes with prolate, spherical, and oblate retinas as functions of field angle. Bilateral differences in magnifications were examined for combinations of those eyes.

**Results**:
Retinal shape substantially affects magnification profiles even for eyes with the same axial length. Greatest magnification changes across a retina and between eyes, as well as greatest increase in radial–tangential differences (distortion), occur with prolate retinas. Binocular magnification differences were smallest for oblate retinas. Nodal points anterior to the vertex center of curvature and oblate asphericity both cause field-dependent reductions in magnification relative to the fovea (barrel distortion), whereas nodal points posterior to vertex center of curvature and prolate asphericity cause the opposite (pincushion distortion). Retinal magnification differences due to eye shape are much greater than aniseikonia thresholds and chromatic differences in magnification. A spreadsheet tool implements the magnification calculations.

**Conclusions**:
Local retinal magnifications as functions of field angle have substantial effects on objective applications (imaging retinal anatomy) and subjective experiences (aniseikonia) and quantify an ocular property that differs across eye shapes and refractive errors.

**Translational Relevance**:
Methods are provided to customize the calculation of radial and tangential magnifications across the retina for individual eyes, which will bolster the multifactorial study of the effects of foveal and peripheral optics across eye shapes and refractive errors.

*radial*magnification relates to the local shape of the retina in a plane (such as a magnetic resonance imaging [MRI] section of an eye

^{1}

^{,}

^{2}), and

*tangential*magnification relates to the distance that a hypothetical ray of light travels to the retina. In a one-dimensional ray diagram, the classical object–image relationship is only illustrated in the radial dimension (Fig. 1B); the tangential dimension is perpendicular to the radial plane (Fig. 1A). Changes in magnification over the visual field and differences between radial and tangential magnifications are what is classically referred to as

*distortion*.

^{3}

^{,}

^{4}or optical coherence tomography,

^{5}

^{,}

^{6}when computing light irradiance across the retina,

^{7}and during focus-dependent surgical techniques such as photocoagulation.

^{8}Ocular magnification also affects subjective visual experience and is considered when minimizing aniseikonia during the prescription of spectacles

^{9}

^{,}

^{10}or intraocular lens powers,

^{11}as well as during low vision care of eyes with central vision loss

^{12}and in understanding the optical and visual quality of emmetropic versus myopic eyes.

^{13}

^{14}is widely used to determine a foveal retinal magnification factor, but this method assumes that the retina and posterior focal planes of the eye coincide, which is the case in neither foveal ametropias nor in peripheral refractive errors. We show in this paper why magnification in and around the fovea is insensitive to retinal shape, and we focus on the more challenging case of calculating retinal magnification factors beyond the fovea.

^{15}driving,

^{16}and mobility,

^{17}as well as in clinical perimetry.

^{18}Further, due to associations between off-axis optics and the onset and progression of myopia,

^{19}peripheral image quality is being increasingly considered in the design of free-form spectacle lenses,

^{20}

^{,}

^{21}custom and orthokeratology contact lenses,

^{22}and head-mounted displays.

^{23}

^{24}and how velocities of retinal images change across different retinal shapes during eye movements.

^{25}Popular methods

^{14}

^{,}

^{26}of projecting object space across the retina assume that the retina is spherical. The shortcoming of this assumption is illustrated in Figure 1B, where circular and elliptical profiles are plotted that represent three anatomically plausible retinal shapes.

^{1}

^{,}

^{27}These three eyes have the same theoretical on-axis axial lengths (24 mm), but, due to their different retinal shapes, magnification as a function of field angle differs substantially across the three retinas (Fig. 1C).

^{26}are not readily customizable for the calculation of magnification in individual eyes. One goal of the present paper was to provide a tool for these calculations that could be implemented in spreadsheet software (see Supplementary Material). Given increased accessibility of technologies that quantify retina shape,

^{1}

^{,}

^{2}

^{,}

^{27}

^{–}

^{29}this tool allows magnification to be calculated over the retina for individual eye shapes analogous to the common presentation of relative peripheral refraction.

^{24}

^{1}

^{,}

^{27}as a circle, prolate ellipse, or oblate ellipse by a radius of curvature

*r*and an aspheric conic constant

*Q*. To demonstrate the calculations, we articulate monocular magnification patterns resulting from the interaction of the three variables and emphasize the effect of retinal shape. Changes in magnification across the retinas are also discussed in the context of classical concepts such as barrel and pincushion distortions, as well as applications such as ocular imaging, aniseikonia, chromatic differences in magnification, and binocular vision. Across these applications, we ask which errors would result (1) if peripheral magnification was assumed to equal magnification at the fovea, and (2) if the retina was assumed to be spherical when it is aspheric? A detailed derivation of the calculations and instructive methods for monitoring anatomical plausibility when performing the calculations are included in the Appendix.

^{5}

^{,}

^{6}

^{,}

^{30}

^{,}

^{31}and have been shown to be in good agreement with real ray tracing over large field angles.

^{32}Using a nodal point convention simplifies the optical contributions of the refracting surfaces and media of an eye into a pair of locations. The classical concept is that a ray that travels at a particular angle toward the first (anterior) nodal point emerges from the optical system in the same direction, along a line that includes the second (posterior) nodal point. The conceptual path through both nodal points and the fovea is known as the visual axis,

^{33}

^{,}

^{34}and field angle is described as the angle formed between a ray and the visual axis (θ in Fig. 1). We use the term

*fixation*to refer to the object location conjugate to the retinal fovea, and

*nodal point*(

*N*′) to refer to the posterior nodal point.

^{35}In this coordinate space, a two-dimensional section of retina can be described

^{1}

^{,}

^{27}using the conic section equation

^{36}

*y*and

*z*are Cartesian coordinates (Figs. 1A, 1B),

*r*is the vertex radius of curvature (at the fovea), and

*Q*is the conic constant that defines oblate (

*Q*> 0; blue in Fig. 1B), spherical (

*Q*= 0; green), prolate (0 >

*Q*> –1; red), parabolic (

*Q*= –1), and hyperbolic (

*Q*< –1) profiles. Note that defining retinal shape with an elliptical conic section in Equation 1 is the general case, within which circular or spherical retinas are included.

^{1}

^{,}

^{2}

^{,}

^{27}

^{–}

^{29}each of which visualizes a different angular extent of retina. The representation used in Figure 1B is similar to MRI studies

^{1}

^{,}

^{2}where a section of the entire eye globe is described by a single ellipse; this approach ensures that the fit ellipse is anatomically plausible. Virtues and limitations of different approaches are considered in the Discussion and Appendix.

*y*-value (immediately before the retina begins to curve back toward the visual axis). To check this plausibility, we begin by calculating axial length from the parameters that define the eye shape:

*r*is measured from the vertex to the center of curvature and is a negative distance. If the calculated axial length (Equation 2) is greater in magnitude than the distance from the nodal point (

*N*′) to the retina, then any ray will reach the retina irrespective of the angle at which it emerges from the nodal point, and magnification calculations do not have to be constrained. However, if the nodal point is farther from the retina than the hypothetical axial length, the maximum valid field angle where a ray from the posterior nodal point will intersect with the retinal asymptote is

*z*is half of the axial length from Equation 2 and

_{geo}*y*is

_{geo}*N*′), which is calculated using classical geometry; and (3) the magnification relative to the geometric center at the ellipse vertex (fovea), calculated as

*z*. The logic behind this calculation is also articulated in the Appendix.

_{geo}^{33}When the retina is spherical and the nodal point is at the retinal center of curvature, local radial and tangential magnifications are constant as a function of field angle. Below, we show that these constant values depend on the radius of curvature of the spherical retina and not on the nodal point position. As the radius of curvature increases (flattens), the (constant) magnification increases. The ratio between (respectively constant) magnifications for different radii of curvature is the ratio between those radii of curvatures.

*r*with the distance from

*N*′ to the retina. As field angle increases, both radial and tangential magnifications change monotonically: decreasing with eccentricity if the nodal point is anterior to the retinal center of curvature (barrel distortion) and increasing with eccentricity when the nodal point is posterior (pincushion distortion). Tangential magnification is always less than radial magnification. At an angle of 90°, local radial magnification equals the constant value determined by the spherical retinal radius of curvature (see above) irrespective of the nodal point position. This is a consequence of circle geometry; a more general solution is provided for any retinal shape below.

*similar*(scaled versions of each other) have the same relative patterns of radial and tangential magnifications as a function of field angle.

*r*; however, for aspheric retinas it is not that simple, and Equation 2 should be used. From Equation 2 one also notes the inverse linear relationship between the conic constant

*Q*and axial length. Recall that the notion of radius of curvature is not as intuitive with aspheric shapes as for spherical retinas, and here it refers to the vertex radius of curvature (that is, at the fovea).

*Q*> –1) causes an increase in local magnification with increasing field angle (pincushion distortion), and oblate asphericity (

*Q*> 0) causes a decrease in local magnification (barrel distortion) (Fig. 3). Tangential magnification is less than radial magnification. If the underlying vertex radius of curvature is changed, all magnification curves shift by a proportional amount.

*r*and

*Q*) irrespective of nodal point position. The dashed horizontal lines (constant magnification) through these convergence points in Figure 4 are empirically determined as the radial magnification values of equivalent spherical retinas with radii of curvature

*r*and

*Q*relate to the aspheric retina and

*r*and

*r*are negative distances. These equivalent spherical retinas intersect the aspheric retinas (in our coordinate space) at the point on the aspheric retinas where

_{es}*z*=

*y*. The magnifications for each

*r*(dashed lines in Fig. 4) are calculated by substituting

_{es}*r*for

_{es}*r*in Equation 6.

^{24}and corrected using spectacles, the nodal point position would be relatively nearer to the retina (lesser magnitudes of

*N*′ in the legend) than if that eye was corrected with contact lenses (or uncorrected) where the nodal point would be relatively nearer to the cornea.

^{37}The magnification curves echo the well-known minification caused by negative-power spectacles compared with contact lenses.

^{38}is 63 mm. Symmetric convergence is modeled at 114 cm from the eyes, which was the median fixation distance for a collection of tasks

^{39}; changing this parameter has a small effect compared to that of eye shape.

_{L}and θ

_{R}) by all points (e.g., point

*x*) in a transverse plane in visual space spanning 400 × 400 mm. For each point in the plane, Figure 5C plots the unsigned difference in field angle (horizontal disparity) subtended at the two eyes (|θ

_{L}– θ

_{R}|). The angular data in Figures 5A, 5B, and 5C apply to all eye shapes. Figures 5D and 5E plot local radial retinal magnification corresponding to the subtended field angles (Figs. 5A, 5B) for left and right prolate retinas like the one in Figure 1. For each point in the plane, Figure 5F plots the difference in magnification between the eyes as a percentage of the left eye values: Positive values indicate magnification relative to the left eye, and negative values indicate minification. Because Figures 5A to 5F model symmetric convergence of identical right and left eyes, there is no difference in magnification at the fixation point (not shown). The difference in magnification between right and left eyes increases the nearer that a point is to the eyes and illustrates, for example, the substantial differences in magnification that the optics of near-eye displays are engineered to overcome by placing the virtual image of the displays farther from the eyes than the actual displays. The application of these magnification methods to virtual reality head-mounted displays is a subset of the free-viewing case illustrated in Figure 5. The virtual image plane is typically at a distance roughly 65 cm to 2 m from the eyes, and accommodation posture—which impacts nodal point position—is expected to be relatively constant around that virtual image plane. Disparity rendering will drive convergence posture—which impacts the field angle relative to the nodal point. Although the optics of high-powered display lenses might distort virtual object space, geometric rendering calibrations should compensate for these distortions, and the three-dimensional space represented in Figure 5 becomes a plane at the virtual image distance from which light originates and projects across the retinas.

^{40}

^{,}

^{41}Whereas Figures 5D to 5F show radial magnification corresponding to the one condition where both eyes are prolate in shape, Figures 5G to 5I illustrate radial and tangential magnifications for many pairings of eye shapes at 25 cm and 10 cm from the eyes. Figures 5G and 5H treat right and left eyes as identical in shape (both prolate or both spherical or both oblate); Figure 5I models right and left eyes that are both prolate but by different amounts and consequently also have different axial lengths. Parameters of that left eye are like the prolate retinas in Figure 1 (

*r*= –9;

*Q*= –0.25; axial length = 24 mm) and parameters of the right eye are

*r*= –11.025,

*Q*= –0.1, and axial length = 24.5 mm. The nodal point of both eyes in Figure 5I was 7 mm from the cornea, mimicking vitreous chamber elongation that is common in anisometropia and aniseikonia.

^{9}

^{9}Neural aniseikonia relates to the relative proportions of the retinas covered by the images, where, in a longer eye, an image of a given size covers a relatively smaller proportion of that retina than the same image size covers in a shorter eye. A simple account of this neural aniseikonia was incorporated into the dashed curves in Figure 5I by multiplying the radial magnification as a function of field angle in one eye by the ratio of the perimeters of the two globes (see Equation A15 and the below section, Extension of Data to Calculate Retinal Arc Length) and using the classical assumption of corresponding retinal points.

^{42}This has only a small effect on the difference in magnification experienced between the two eyes (Fig. 5I); the majority is caused by the differences in optical magnification described throughout this paper.

^{34}and have been shown to be a good approximation of real ray tracing over wide field angles.

^{32}Despite that, at large field angles, the ray path through the entrance pupil (line of sight) can differ from the path modeled through the nodal points (visual axis),

^{43}

^{,}

^{44}we treat this difference as negligible, given that the main effects of the magnification results arise at angles larger than 5°. Although being a classic concept in optics, nodal points maintain modern applications such as in widefield

^{6}and ultra-widefield

^{5}optical coherence tomography, the design and troubleshooting of intraocular lens complications (such as farfield negative dysphotopsia

^{31}), determining posterior vitreous chamber dimensions,

^{30}modeling peripheral field loss,

^{45}and studying retinal structure–function relations dependent on field eccentricity.

^{18}

^{26}

^{,}

^{46}and instrumentation assumptions.

^{47}Multiple nodal point positions were modeled to illustrate the suitability of these methods for applications where the nodal point position changes, such as across different refractive errors

^{37}and during accommodation.

^{46}Nodal point positions also change when correcting ametropia with spectacle or trial lenses. Compensating hyperopia with a positive lens moves the nodal point anteriorly (toward the cornea), whereas compensating myopia with negative-powered spectacles moves the nodal points posteriorly (toward the retina).

^{48}

^{,}

^{49}Although changes to the nodal point position are minimally impacted by contact lens corrections

^{37}and can be completely avoided by using a Badal optometer, the compensation of ametropia with spectacle or trial lenses might complicate other imaging applications where assumptions have been made about nodal point positions. Nodal point positions for these magnification calculations either can be sensibly selected from the above-mentioned literature or can be estimated using customized or semi-customized biometric eye models in optical design software,

^{3}

^{,}

^{4}which can also include any corrective lenses.

^{27}of retina could be more accurately fit, those methods can result in a larger range of values across individuals and implausible axial lengths if the fit parameters of the retinal segment are extrapolated to describe the entire globe. By beginning with a fit of the whole globe,

^{1}

^{,}

^{2}we are able to estimate and evaluate the anatomical plausibility of axial length and limit field angles to those that will reach the retina (Equations 2–4).

^{14}

^{,}

^{26}

^{14}rather than projected across the curved retinal surface. In reality, different retinal shapes also result in different locations where a ray will intersect with (form an image on) the retina. The length along the retina from the fovea to the retinal image could be calculated using Equation A15 (perimeter of an ellipse) by modifying the integral (and coefficient factor) from π/2 to the desired field angle; it could be equivalently calculated (as was done for validation of this subanalysis) using elliptic integrals.

^{50}Alternatively, that length along the retina from the fovea (θ = 0°) to the retinal image for a ray emerging from the nodal point at θ =

*h*° is the cumulative sum of magnifications (mm/deg) for angles from 0° to

*h*° (essentially a discrete integral). Figure 6B plots retinal images of a 1°-diameter circle at the locations where they form on the retinas. One can appreciate how visual space is minified across the prolate retina relative to the oblate retina.

^{51}

^{,}

^{52}Although differing spectral sensitivities play a role, the percentage of chromatic magnification differences with which the visual system is equipped to deal is substantially smaller than the effects demonstrated here from different retinal profiles.

^{53}

^{54}which allows modification of monocular optical magnification to balance aniseikonia. Foveal data are plentiful and generally in agreement that typical levels of aniseikonia are less than a 2% difference in image size between the eyes and that a substantial impairment of binocular vision occurs at approximately 3% to 5% difference.

^{10}

^{,}

^{55}

^{56}summarized three studies and concluded that Panum's area (of single vision) grows relatively smaller with peripheral eccentricity. Crone and Leuridan

^{53}measured diplopia thresholds in degrees (asking whether two line targets appeared merged as single or distinct as double) and found a gradual increase in those diplopia thresholds as field angle increased beyond 10°. They believed that aniseikonia thresholds will also increase with field angle. Magnification differences between right and left eyes (Fig. 5) are substantially dependent on the distance of a point or plane from the eyes. For the examples presented, objects 25 cm from the eyes (Fig. 5G) resulted in magnification differences similar to the thresholds of Ames and Ogle

^{56}and Crone and Leuridan.

^{53}For objects nearer to the eyes (Figs. 5F and 5H) binocular magnification differences increased substantially and varied dependent on eye shape: In prolate retinas, these were considerably larger than the thresholds from the literature,

^{53}

^{,}

^{56}but oblate retinas shapes mitigated binocular magnification differences (Fig. 5).

^{1}

^{,}

^{2}

^{,}

^{27}

^{–}

^{29}can be input or a selection of retinal shapes can be iteratively modeled, and local radial and tangential retinal magnifications as functions of field angle can be calculated. Again, nodal point position can be selected from the known assumptions of instrumentation,

^{47}literature,

^{37}

^{,}

^{46}generic eye models,

^{26}

^{,}

^{46}or custom eye models

^{3}

^{,}

^{4}and optical design software. The effects of retinal shape on local magnification across the retina are substantial and are important to consider in objective applications such as imaging retinal structures, subjective patient experiences such as aniseikonia and diplopia, and as an additional ocular property to bolster the multifactorial study of the effect of optics across eye shapes and refractive errors.

**G.D. Hastings**, None;

**M.S. Banks**, None;

**A. Roorda**, None

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^{35}In this coordinate space, a two-dimensional section of retina can be described

^{1}

^{,}

^{27}using the conic section equation:

^{36}

*y*and z are coordinates in a Cartesian sense (Fig. 1),

*r*is the vertex radius of curvature (that is, radius of curvature of the retina at the fovea), and

*Q*is the conic constant that defines oblate (

*Q*> 0; blue in Fig. 1B), spherical (

*Q*= 0; green), prolate (0 >

*Q*> –1; red), parabolic (

*Q*= –1), and hyperbolic (

*Q*< –1) profiles. Note that defining retinal shape with an elliptical conic section in Equation A1 is the general case, within which the specific case of a circular or spherical retina is included. Literature

^{36}contains an alternate definition of the conic constant using the symbol

*P*, where

*P*=

*Q*+ 1.

*C*is the retinal vertex curvature and

*C*= 1/

*r*.

*N*′) at angle θ and traveling toward the retina (such as in Fig. 1B) using the familiar straight-line equation:

*y*-intercept of the straight line is

^{1}

^{,}

^{2}

^{,}

^{27}

^{–}

^{29}each of which visualizes a different angular extent of retina. The representation in Figure 1B is similar to MRI studies

^{1}

^{,}

^{2}where a section through the entire eye globe is described by a single ellipse. Although fitting smaller angular portions of the retina

^{27}might provide more accurate fitting over those segments, if the fit parameters are extrapolated to describe an entire eye globe it opens the possibility of anatomically implausible eye shapes. For example, if a posterior section of retina is best fit by a parabolic conic section, the anterior part of that hypothetical eye will never close at all, and axial length (Equations 2 and A6) in the magnification calculations is undefined. Alternatively—at the other extreme of asphericity—if a segment of posterior retina is described by a substantially oblate ellipse, it is possible that the extrapolated full ellipse will be too short to plausibly define an anatomically typically axial length. Therefore, if only a segment of the retina has been fit, it is prudent to determine whether an elliptical fit to that segment can be meaningfully extended to define an entire eye. To check this plausibility, we begin by calculating axial length from the parameters that define the eye shape:

*r*is measured from the vertex to the center of curvature and is a negative distance. For segments of posterior retinal profiles that are fit with parabolic ellipses (

*Q*= –1) (and will not converge to form an anterior portion of an eye), the denominator of Equation A6 becomes zero and axial length is undefined.

*N*′) to the retina, then any ray will reach the retina irrespective of the angle at which it emerges from the nodal point, and the magnification calculations do not need to be restricted. However, if only a segment of retina has been fit with an oblate ellipse and those fit parameters are extrapolated to define an entire eyeball, it is possible that the nodal point may be farther from the fovea than the hypothetical axial length (Fig. A1). In such cases, we limit that calculation of magnification to the maximum

*y*-value (immediately before the retina begins to curve back towards the visual axis). For any ellipse (or circle), the

*z*-value at which that happens is the midpoint of the theoretical axial length, which corresponds to the

*z*-value of the geometric center (

*z*) of the ellipse and half of the value determined by Equation A6 (illustrated in Fig. A1). Here,

_{geo}*z*is a negative distance relative to the fovea. The maximum

_{geo}*y*-value (

*y*) can be calculated by substituting

_{geo}*z*into Equation A1 and solving for

_{geo}*y*. Alternatively, a simpler calculation of

*y*is possible by noting that the point where the retinal shape begins curving back toward the visual axis is when the square root in the denominator of Equation A2 is not real; thus, when

_{geo}*y*-value) is when

*y*is positive and

_{geo}*z*and

_{geo}*N*′ are negative.

*k*(

*z*) on the retina of a ray emerging from the nodal point at angle θ. This is accomplished by formulating Equations A1 and A3 in terms of

_{int},y_{int}*y*, setting them equal to each other, and solving for the roots of

*z*using the classical

*quadratic formula*:

*z*from the general quadratic formula are

*A*and

*B*are the coefficients of

*z*and

^{2}*z*, respectively, and

*C*is the constant term; that is,

*y*-coordinates corresponding to each

*z*-root are calculated by substituting each

*z*(Equation A13) into either Equation A1 or Equation A3. Because θ is measured counterclockwise, we select the

_{r}*z*-root that corresponds to a positive

*y*-coordinate.

*z*,

_{int}*y*) at the nodal point is θ; in the next section, we will require the angle (ɸ) subtended by the same retinal point at the geometric center of the ellipse (Fig. 1B). The quadrant-specific arctangent formulation is

_{int}*y*is positive and

_{int}*z*and

_{int}*z*are negative.

_{geo}^{57}Calculation of tangential magnification will follow from the radial method.

*ɑ*is the length of the semi-major axis of the ellipse, ɸ is the angle subtended at the geometric center of the ellipse (not at the nodal point; see Fig. 1B), and

*e*is the ellipse eccentricity defined as

*b*is the length of the semi-minor axis of the ellipse. For the spherical or circular case,

*ɑ*=

*b*and

*e*= 0. For all retinal shapes, we have already calculated magnitudes of

*ɑ*and

*b*under other names above:

*similar*ellipses can all have identical eccentricity. Also note that

*ɑ*is always for the major (longer) axis and

*b*is always the minor (shorter) axis; therefore, ellipse eccentricity is insensitive to prolate or oblate orientations (conic constant

*Q*).

*e*.

*N*′), (3) and the magnification relative to the geometric center at the ellipse vertex (fovea). These methods are automated in a spreadsheet in the Supplementary Materials. First, local relative magnification is from Equation A20. Second, if an angle subtending 1° is projected from the nodal point and centered around the ray at the angle of interest θ, then the angle subtended at the geometric center is calculated using Equations A3, A4, A5, A12, A13, and A14. Third, when the field angle (θ) is zero, magnification is determined by the distance along the visual axis to the vertex. Here, that distance is from the geometric center and

*z*-axis (Fig. 1A). Tangential magnification is ultimately calculated using Equation A21 and substituting the path length of the ray (from the nodal point to the retina) in the place of

*z*. Briefly, we articulate the logic leading to that conclusion.

_{geo}*k*) can be defined as the new origin of this new two-dimensional space. Having an ellipse with the vertex at the new origin of this coordinate space, we can again apply the magnification method described above. Because instantaneous tangential magnification relates to the path length of ray from the nodal point to the retina, tangential magnification is determined by substituting that path length into Equation A21.

^{58}good agreement was found between Zemax, which defines and traces rays with reference to the entrance pupil, and the present method that uses the nodal point, echoing other widefield findings.

^{32}