New results in stereopsis and Listing’s law

Abstract

Human eyes’ optical components are misaligned. This study presents comprehensive geometric constructions in the binocular system, with the eye model incorporating the fovea that is displaced from the posterior pole and the lens that is tilted away from the eye’s optical axis. It extends their previously considered horizontal misalignment with the vertical components. When the eyes’ binocular posture changes, 3D spatial coordinates of the retinal disparity (iso-disparity curves), the subjective vertical horopter, and the eye’s torsional orientation transformations are visualized in GeoGebra’s simulations. The consequences and functional roles of vertical misalignment of the eye’s optical components are explained in the following findings: (1) The classic Helmholtz theory, which states that the subjective vertical retinal meridian inclination to the retinal horizon explains the backward tilt of the perceived vertical horopter, is less relevant when the eye’s optical components are misaligned. Instead, the lens vertical tilt provides the retinal vertical criterion that explains the experimentally measured vertical horopter inclination. (2) Listing’s law, which originally restricts single-eye torsional positions and has imprecise binocular extensions, is formulated for binocular fixations using Euler’s rotation theorem. It, however, replaces Listing’s plane, which is defined for eyes looking at infinity, with the eyes muscles’ natural tonus resting position corresponding to the abathic distance fixation of empirical straight frontal horopter. This new meaning of Listing’s plane provides neurophysiological significance that has remained elusive.

Introduction

The optical components of the healthy human eyes are misaligned; the fovea is not located on, and the crystalline lens is tilted away from the eye’s optical axis. This asymmetry has been accounted for in many clinical studies, usually decomposed along temporal-nasal and inferior-superior axes, that is, into horizontal and vertical segments, respectively^1,2,3,4,5.

The consequences of the horizontal misalignment of the eye’s optical components for the quality of vision are well known—the fovea’s anatomical displacement in the retina from the posterior pole contributes to optical aberrations, and the lens tilt partially compensates for these aberrations^6,7,8,9.

It is reasonable to expect the horizontal misalignment to impact the foveal correspondence. For a small retinal area in one eye, there is a corresponding unique area in the other eye such that both share one subjective visual direction. This horizontal correspondence between disparate 2D retinal images of our laterally separated two eyes organizes visual perception of objects’ form and their location in 3D space, i.e., stereopsis and spatial relations^10,11. The impact of eye horizontally misaligned optical components on stereopsis and visual space geometry was confirmed in the geometric modeling in^12,13,14 of the binocular system with the asymmetric eye (AE) model that comprised the horizontal misalignment of the fovea and the crystalline lens. The results of those studies are briefly reviewed later in this section, which should help read the rest of the paper.

The study presented here provides the comprehensive geometric construction of spatial coordinates of the retinal disparity correspondence by including the fovea’s and lens’ vertical misalignment in the AE model. These spatial coordinates represent the apparent frontal lines as iso-disparity conic sections, where the zero-disparity curve is the longitudinal horopter and the apparent vertical direction at the point of fixation, which should be identified with the vertical horopter. Their transformations are visualized in GeoGebra’s dynamic geometry simulations between different binocular postures. Further, torsional disparity for eyes’ postures is computed using Euler’s rotation theorem in Rodrigues’ (rotation) vector framework and visualized in GeoGebra’s simulation. It provides a consistent binocular formulation of Listing’s law for the first time. Table 1 summarizes the notation and important geometrical facts used in this paper.

The results of this study provide new ideas for the physiologically motivated geometric conceptualization of binocular vision. In particular, they explain the functional role of vertical misalignment of the optical components as the new criterion of retinal vertical correspondence, which explains the subjective vertical horopter inclination from true vertical observed in experiments.

Table 1 Summary of the notation and important definitions used in the text. The subscripts ‘r’ and ‘l’ indicate the right and left eye.

Prior studies with the AE

The first asymmetric eye (AE) constructed in¹² comprised the fovea’s horizontal displacement from the posterior pole by angle $\alpha =5.2^{\circ }$ and the crystalline lens’ optical axis horizontal tilt relative to the eye’s optical axis by angle $\beta$. Angle $\alpha$ is relatively stable in the human population¹⁵ and angle $\beta$ predominantly varies between $-0.4^{\circ }$ and $4.7^{\circ }$. The value $\beta =3.3^{\circ }$ corresponds to the average abathic distance fixation of about 1 m for the empirical horopter—the unique fixation for which the horopter is a straight frontal line. In Fig. 1, (a) presents the standard single refractive surface schematic eye, (b) the AE model, and (c) the abathic distance fixation in the binocular system with the AEs discussed below. This figure also establishes the basic notation used in the AE model and its important properties.

Fig. 1

(a) The axially symmetric eye’s model of a single refractive surface schematic eye. (b) The right (2D) AE. N is the nodal point, C is the eye’s rotation center, f is the fovea, and p is the posterior pole. The image plane is parallel to the lens’s equatorial plane and passes through the eye’s rotation center. O is the projection of the fovea into the image plane—the optical center on the image plane. The lens is represented by the image plane and the nodal point. (c) The binocular system with the AE model in the ERP is seen from above. The fixation $F_{a}$ corresponds to the AE’s posture in which the coplanar equatorial lens planes are parallel to the coplanar image planes. Under these conditions, the horopter through $F_a$ is a straight frontal line resembling the empirical horopter at the abathic distance. The point Q on the horopter projects through the nodal points $N_{r}$ and $N_{l}$ to the corresponding retinal points $q_{r}$ and $q_{l}$ and to points $Q_{r}$ and $Q_{l}$ in the image plane. The fixation $F_a$ projects to the foveae $f_r$ and $f_l$ in the retina and the optical centers $O_r$ and $O_l$ in the image plane. Although $|f_{l}q_{l}| \ne |f_{r}q_{r}|$ on the retina, $|O_{l}Q_{l}|=|O_{r}Q_{r}|$ on the image plane.

In the geometric theory developed in¹³, the retinal correspondence of the binocular system with AEs is first constructed for the posture corresponding to the abathic distance fixation. In this posture shown in Fig. 1(c), the coplanar eyes’ image planes are parallel to the coplanar lens’s equatorial planes, which determines the distance to the fixation point $F_a$,

$$\begin{aligned} |OF_{a}|=\frac{a\cos (\alpha -\beta )+0.6\sin \alpha }{\sin (\alpha -\beta )}, \end{aligned}$$

(1)

where $2a=6.5$ cm is the ocular separation, and 0.6 cm is the distance of the nodal point from the eyeball’s rotation center. With $a=3.25$ cm, $\alpha =5.2^{\circ }$ and $\beta =3.3^{\circ }$, $|OF_a|=99.6$ cm.

This binocular posture is referred to as the eyes’ resting posture (ERP) because it numerically corresponds to the eyes’ resting vergence posture of the fixation about 1 m, in which the eye muscles’ natural tonus resting position serves as a zero-reference level for convergence effort¹⁶. As explained in the caption of Fig. 1(c), the result is the symmetric distribution of the points on the intersection line of the visual plane and image planes of the AEs. Thus, the unknown asymmetric retinal correspondence is expressed in AE’s angles $\alpha$ and $\beta$ under its projection onto the image plane through the corresponding nodal point.

For all other fixations in the horizontal plane, the horopters simulated in GeoGebra’s dynamic geometry environment consist of ellipses or hyperbolas resembling the empirical horopters. Importantly, the correspondence symmetric distribution on the image plane is preserved for all of these fixations, providing the first evidence that the asymmetry of retinal corresponding elements is caused primarily by the horizontal misalignment of the eye’s optical components. The retinal correspondence asymmetry determines the shape of the longitudinal empirical horopter^17,18,19.

Further, in¹⁴, the zero-disparity horopter is extended to the families of iso-disparity conic sections, visualized in GeoGebra’s simulations and applied to study in the framework of Riemannian geometry the global aspects of phenomenal spatial relations—visual space variable curvature and finite horizon. The global aspects of stereopsis are crucial in perception, as exemplified by the coarse disparity contribution to our impression of being immersed in the ambient environment despite receiving 2D projections of the light beams reflected by spatial objects²⁰.

Importantly, the GeoGebra simulations in¹⁴ revealed that the distribution of iso-disparity lines in the ERP is invariant to changes of the AE parameters $\alpha$ and $\beta$. This invariance, demonstrated in that reference in Figs. 2 and 5, shows that the distribution of iso-disparity lines is universal. For any parameters $\alpha$ and $\beta$ of the AE that uniquely prescribe the ERP in (1), iso-disparity lines agree with this universal distribution. Moreover, the distance between iso-disparity lines can be scaled down to the cone photoreceptors separation in the foveal center; see Fig. 6 in¹⁴.

The results described in this section firmly establish for the first time that the human eye misalignment of optical components results in the asymmetry of retinal corresponding elements. This asymmetry, in turn, establishes the shape of the iso-disparity variation with the eyes’ binocular fixation as the conic sections that resemble empirical horopter variation.

Many different computational techniques have been applied to study stereopsis. Some were based on salient features such as edges or corners²¹, and others were based on higher-order properties of surfaces interpolated from data points to obtain the depth map, see²² for example. Still, others lead to the orientation, spatial frequency, or gradient disparities^{23,24,25,26,27}. Despite the contributions from these approaches to stereopsis, the recent claim in²⁸ that retinal corresponding elements and disparity spatial coordinates (iso-disparity curves) are not necessary for stereoscopic vision is unconvincing as it is demonstrated by results obtained in my studies that were described above.

3D AE and the criterion of retinal vertical correspondence

Based on the clinical studies listed in the first section, the angles of $\gamma =-2^{\circ }$, $\varepsilon =-1^{\circ }$ are assumed for the vertical asymmetry of the fovea and the lens, respectively, complementing $\alpha =5.2^{\circ }$ and $\beta =3.3^{\circ }$ angles of horizontal asymmetry. For these asymmetry angles, the lens equatorial planes and, hence, the image planes of the AEs are coplanar for the fixation $F_a(0,99.56,1.72)$ expressed in centimeters in the upright head coordinate system—this is the ERP. Figure 2 shows the ERP of the binocular system perspective view of the right 3D AE near the optical center $O_{r}$ and the eye’s rotation center $C_r$ in the image plane. The coordinates of $O_r$ and $N_r$ are obtained in GeoGebra’s simulation for the values of the misalignment angles. The left AE is the mirror-symmetric in the median plane.

Fig. 2

The right 3D AE in the ERP. The yellow line is the eye’s optical axis, the red line is the visual axis fixating $F_a$, and the blue line is the lens’s optical axis. The AE’s image plane coordinates consist of the axis $H_r$ of the intersection by the visual axis and the vertical axis $V_r$, both attached at the optical center $O_r$. They are complemented by the perpendicular coordinate $T_r$ parallel to the lens’s optical axis. These three axes in the ERP are parallel to the upright head coordinate axes. The other details of this figure are discussed in the text.

This figure also explains the definitions of the AE’s misalignment angles and their impact on the geometry of the binocular system. The axis $H_r$ is the axis of retinal horizontal correspondence—the location of the corresponding points on the intersection of the coplanar image planes parallel to the lenses’ equatorial plane with the visual plane. Thus, the line $H_r$ is usually called the foveal horizon. The inclination of the visual plane passing through $O_r$, $O_l$ and $F_a$ is $\eta =\arctan (1.741/99.56)=1^{\circ }$. The direction of retinal vertical correspondence $\mathcal {V}_r$ differs from the vertical axis $V_r$ by the angle $\iota =0.183^{\circ }$ constructed below for the AE’s misalignment angles. The coordinate axes $H_r$ and $\mathcal {V}_r$ that express the horizontal and vertical correspondence in the image plane are shown in Fig. 2 in green.

Helmholtz’s classic theory²⁹ asses that in the binocular system fixating on a point in the median plane, the meridians of the vertical retinal correspondence are not perpendicular to the retinal horizon. This lack of perpendicularity is known as Helmholtz’s shear. The objection to Helmholtz’s²⁹ theory, and the later studies is that the main vertical meridians are not well defined in the human eyes because of the global eyeball asymmetry caused by the fovea displacement on the retina from the posterior pole by about $5^{\circ }$ measured at $N_r$. In the geometric theory of the binocular system with AEs, the meridian criterion is replaced with the lens orientation criterion of verticality.

Before I continue, however, I recall the main postulates underlying geometric theory in^13,14. In the AE, the lens is represented by the nodal point and the image plane that is parallel to the lens’s equatorial plane. Then, the unknown asymmetric retinal correspondence is represented by its projection through the nodal points into the image planes of the AEs. The resulting image plane correspondence, first constructed for the ERP, is symmetric relative to the optical center—the projections of the foveae; cf. Fig. 1(c). Because in that theory, the eyes in the upright head are binocularly fixated on points in the horizontal plane, the image plane corresponding points are always located on the line intersecting the visual plane and the image plane, which is precisely the foveal horizon of the binocular system.

In this study, however, the eyes can fixate on the points of the 3D binocular field. Thus, in tertiary eyes postures, the ocular torsion twists the retina about the visual axis, resulting in nonzero torsional disparity. However, for a constant angular size of this disparity, its linear size is increasing with eccentricity. The binocular single vision is maintained in the peripheral region by sensory cyclofusion of up to $8^{\circ }$ supported by peripheral large receptive fields³⁰. Although one can measure fusion horopter^31,32,33 to determine the horopter curve as its medial axis, it will not help my geometric modeling of stereopsis.

Here, the horizontal correspondence is again expressed on the intersection line of the visual plane and the image plane of the AE, initially in the ERP and then maintained in simulations for other fixations. This simple construction is allowed by the sensory cyclofusion range of $8^{\circ }$ in addition to the active compensation for the ocular torsion by the brain processing³⁴ and the fact that the largest torsional disparity in the simulations is less than $1^{\circ }$.

Because the main vertical meridian that serves as the basis for the verticality criterion in Helmholtz theory is not well defined in the eye with the misaligned optical components, the vertical line in the equatorial plane of the lens substitutes the main meridian. This line tilts when the lens undergoes both horizontal and vertical tilt and is always parallel to the image plane, defining the direction of vertical retinal correspondence, c.f. Fig. 2.

The line of vertical retinal correspondence is constructed as follows. First, I denote by $M_r$ the plane containing $O_H$, $n_r$, and $N_r$, where $n_r$ is the orthogonal projection of $N_r$ into the image plane and the $O_H$ is the origin of the upright head coordinate system. The normal vector to $M_r$ is parallel to the image plane and is taken as the direction vector of the green line passing through $N_r$ in Fig. 2. The $M_r$ normal vector, which is perpendicular to the position vector $\textbf{n}_r$, is given by $\textbf{v}=(0.0105,0,3.2845)$. It defines the line of vertical foveal correspondence $\mathcal {V}_r$ passing through $O_r$ (shown in Fig. 2 as the green line) and tilted by $\iota =\arctan (0.0105/3.2845)=0.183^{\circ }$ relative to the objective vertical axes $V_r$ with its top in the temporal direction. Because of the mirror symmetry in the medial plane in the ERP, the left AE’s vertical retinal correspondence is similarly formulated. Thus, the tilt between the lines of vertical retinal correspondence is $2\iota =0.366^{\circ }$.

I end this section by presenting in Table 2 results for a couple of different vertical misalignment values of $\gamma$ and $\varepsilon$ for the constant values of $\alpha$ and $\beta$ listed above. The distribution of iso-disparity curves was studied for three different values of $\beta$ in¹⁴. The table data include the corresponding fixation point $F_a$, the tilt of the retinal apparent vertical $\iota$, and the inclination of the visual plane $\eta$.

Table 2 Results for the different values of AE’s vertical angles $\gamma$ and $\varepsilon$ for $\alpha =5.2^{\circ }$ and $\beta =3.3^{\circ }$.

The results obtained in this section and summarized in Table 2 should be noted for their simplicity. First, the retinal apparent vertical tilt increases by $0.183^{\circ }$ for each additional $-1^{\circ }$ of the lens vertical tilt $\varepsilon$. Second, the inclination of the visual plane $\eta$ is equal to $\varepsilon -\gamma$.

Method: GeoGebra simulations and geometric constructions

This section discusses how iso-disparity curves and vertical horopter are constructed in GeoGebra simulations and how the horizontal, vertical, and torsional disparities are calculated.

GeoGebra simulations

GeoGebra simulations of the eyes’ binocular postures include geometric primitives (conics, spheres, cylinders, etc.), geometric constructs (tangent, bisector, etc.), and basic geometric transformations (translations, rotations, etc.). These geometric tools are linked such that by moving the fixation point, the binocular system with AEs accordingly changes, visualizing its transformations. Thus, programming is not involved. In the simulations presented in this study, the number of geometric tools is quickly growing, and those that create connections allowing geometric transformations are hidden. The large number of hidden geometric objects makes simulation accessible but prevents a deeper understanding of the geometric constructions.

Because the GeoGebra does not involve programming, the links to simulations of iso-disparity conic sections, vertical horopter, and ocular torsion in the binocular system with AEs including detailed instructions are available upon reasonable request.

Iso-disparity conic sections and the vertical horopter

Figure 3 explains the construction of disparity conic sections and the vertical horopter using the right eye for the fixation $F_a(0, 99.56,1.72)$ at the ERP. The coordinates are in centimeters.

Fig. 3

The construction of the iso-disparity and vertical horopter is shown for the right eye of the ERP. The red line passing through the nodal point $N_r$ and the optical center $O_r$ is the visual axis, and the yellow line from $C_r$ and through $N_r$ is the optical axis. The blue line through $N_r$ and $n_r$ is the lens’s optical axis perpendicular to the image plane. The fixation point is F(0, 99.56, 1.72), and the horizontal and vertical coordinates relative to $O_r$ are listed. The green line $H_r$ through $O_r$ is the foveal horizon, with the horizontal disparity points placed 0.005 cm apart. The green line $\mathcal {V}_r$ through $O_r$ is the axis of vertical retinal correspondence.

The iso-disparity lines in physical space are first constructed for ERP. In Fig. 3, the points $Q_{r}=O_{r}\pm n\delta$, where $\delta =0.005$ cm and $n=1,2,..$, are shown on the foveal horizon line. The n-th iso-disparity line is obtained for all pairs $(Q_r, Q_l)$ such that $Q_r-Q_l=n\delta$. For n and $n+1$, the consecutive straight frontal iso-disparity lines for the ERP of fixation point $F_a(0,99.56,1.72)$; c.f. Fig. 5 of the simulation. Then, the iso-disparity curves are simulated in GeoGebra by changing $F_a$ to other fixations.

To construct the subjective vertical horopter, I choose two points $V_u$ and $V_d$ on the apparent vertical line $\mathcal {V}_r$ in the image plane passing through the optical center $O_{r}$ shown in Fig. 3 for the right AE. I recall that this line is the axis inclined $0.183^{\circ }$ top in the temporal direction from the vertical. It indicates the foveal vertical criterion specified by the 3D lens tilt. The points $V_u$ and $V_d$ are the same distance from the optical center. The same construction is carried out for the left AE. Two intersections of the projecting rays, one for $V_u$ points and the other for $V_d$ points of the two AEs would define the vertical horopter. However, the intersection is not empty for fixations in the median plane.

The vertical horopter is extended to the fixation’s central range of azimuthal angles by modeling the 3D property of this horopter, first emphasized by Amigo in³¹, who studied the stereoscopic sensitivities of the retinal regions at the various elevations above and below the fovea. The vertical horopter’s extension for asymmetric fixations is modeled as follows. The two back-projected rays, one for each AE used for constructing the vertical horopter in the ERP, are replaced with cylinders. The other two rays intersect the cylinders, and the vertical horopter passes through the midpoints of the intersections. The cylinder’s radius determines the vertical horopter range of the azimuthal angles. In the simulations, I assumed a 1 cm radius at the abathic distance of about 100 cm and 30 cm above and below the fixation point.

Ocular torsion and the AE

Ocular torsion is usually described as a rotation in the eyeball around the line of gaze. This definition is impractical for the AE because it includes misaligned optical components. The geometrical ocular torsion of the AE is defined as a rotation around the image plane perpendicular coordinate axis at the optical center—the axis $T_r$ for the right AE in Fig. 2. Because this axis is parallel to the lens’ optical axis at a distance of 0.022 cm, ocular torsion around this axis is the closest to rolling about the corneal axis. How the ocular torsion is computed and simulated in GeoGebra is explained with the help of Fig. 4 and discussed later for the actual simulation. In this figure, the small translation components of $O_r$ and $n_r$ when the eye is rotated at $c_r$ (cf. Fig. 2) do not impact ocular torsion and are not shown for clarity. Later, translations of $O_r$ and $O_l$ with the change of the fixation point, which are important in simulation of the half-angle rule, are included.

Fig. 4

Schematic explanation of how the ocular torsion is geometrically defined and calculated for the right eye. The black frame at $O_r$ agrees with the axes $H_r, T_r, V_r$ in Fig. 2 and with the head coordinates x, y, z. The green frame is the transformed black frame after the posture changed from the fixation at $F_a$ in the ERP to a fixation F. The green ‘primed’ frame in dashed lines shows the rotated green frame in the plane containing the vectors $\textbf{b}_r$ and $\textbf{j}_r$ that overlays $\textbf{b}_r$ with $\textbf{j}_r$. The geometric definition of the ocular torsion is the angle $\tau _r=\angle (\textbf{k}_r,\textbf{c}'_r)$. By the definition of ocular torsion, the rotation vector of the black frame is contained in the image plane of the ERP, and ocular torsion is about the line through $O_r$ parallel to the lens’s optical axis (the line through $n_r$ and $N_r$) which is perpendicular to the image plane.

The results establish the geometrical calculations of the eyes’ ocular torsions for any binocular fixation executed from the ERP. The binocular ERP provides a neurophysiologically motivated new definition for Listing’s law eye’s primary position originally considered for the single eye; see discussion in¹³. Section “Listing’s law and the AE: Rodrigues’ vector computational framework discusses the relation between the ocular torsion calculated here and by Listing’s law, commonly used to control 3D eye orientations.

Simulations of retinal correspondence spatial coordinates

In this section, I visualize in GeoGebra the longitudinal disparity spatial coordinates (iso-disparity curves) and the subjective vertical horopter in the binocular system with 3D AE for the primary, secondary, and tertiary postures.

The iso-disparity curves and the subjective vertical horopter are precisely defined in the binocular system with AEs only for fixations in the median plane. Sections “3D AE and the criterion of retinal vertical correspondence” and GeoGebra simulations discussed the extensions of these curves and the vertical horopter.

Disparity spatial coordinates for ERP

The fixation point $F_{a}$ of the ERP of the stationary upright head is uniquely determined by the frontal orientation of coplanar image planes. Then, $F_{a}$’s coordinates in centimeters are (0, 99.56, 1.72), corresponding to the average abathic distance of empirical horopters of about 1 meter. The simulated iso-disparity conic sections in the visual plane are straight frontal lines in the ERP. Also, in this posture, the vertical horopter is a straight line tilted from the true vertical by $\kappa =5.5^{\circ }$ with its top away from the head, shown in Fig. 5.

Fig. 5

The ERP with the fixation at $F_a$ showing the iso-disparity frontal lines in the visual plane and the vertical horopter tilted by $\kappa =5.5^{\circ }$ relative to the objective vertical. The longitudinal and vertical horopters passing through $F_a$ are shown in red. The point $F_0$ is the projection of $F_a$ into the head’s transverse plane containing the eyes’ rotation centers (marked with the grid). The other details of the figure are discussed in the text.

This figure shows the disparity lines for the proximal relative disparity of $\delta =0.005$ cm (see Fig. 3). The point P in the binocular field, shown in Fig. 5, is projected into the AE image planes. The right image plane projection is $P_r(d_{rh}(P),d_{rv}(P))$ and the left image plane projection is $P_l(d_{lh}(P),d_{lv}(P))$ in the coordinates $H_r$ and $\mathcal {V}_r$ and $H_l$ and $\mathcal {V}_l$, respectively. From the simulation computed in GeoGebra and displayed in Fig. 3 for he right AE, the horizontal disparity of P is

$$\begin{aligned} \delta (P)_{h}=d_{rh}(P)-d_{lh}(P)=0.231-0.211=0.02 \, \text {cm}, \end{aligned}$$

(2)

but its vertical coordinates $d_{rv}$ and $d_{lv}$ are the same, so $\delta (P)_v=0$.

The projection of P into the visual plane along the line parallel to the vertical horopter, $P_a$, is exactly on the fourth consecutive disparity line down from the horopter (the horizontal red frontal line through $F_a$). Because the horizontal retinal relative disparity value between each consecutive disparity line is 0.005 cm, the spatial disparity of P agrees with the retinal horizontal disparity $\delta (P)_h=4(0.005)=0.02$ cm. It indicates that the perceived vertical and the vertical horopter should be identified. The horizontal disparity for crossed disparities is always considered positive, and for uncrossed disparities, it is negative.

The ocular torsion, $\tau _r$, and $\tau _l$, computed in this simulation, is zero, such that the torsional disparity $\delta _t$ vanishes in this posture. For details on ocular torsion simulation, refer to Sections “Simulations of retinal correspondence spatial coordinates” and “Simulation of ocular torsion”.

Disparity spatial coordinates for eyes secondary postures

The iso-disparity curves and the vertical horopter are shown in Fig. 6 for the fixation point F that is shifted from ERP by the down-shift from $F_{a}(0,99.56,1.72)$ to $F(0, 99.56, -21.72)$.

Fig. 6

Under the true vertical shift of the fixation point from the ERP, the iso-disparity lines move with the visual plane and remain straight frontal lines. The subjective vertical horopter is tilted from the true vertical direction by $\kappa =18.9^{\circ }$. See the text for a detailed discussion.

This figure shows that the iso-disparity lines move with the visual plane remaining straight and frontal. Further, the subjective vertical horopter is tilted in the midsagittal plane top-away by $\kappa =18.9^{\circ }$ at the fixation point from the objective vertical line shown by a dashed line through $F_a$.

Under this down-shift movement of $F_a$, the horizontal disparity of point P shown in this figure (this point is different than point P in Fig. 5) is

$$\begin{aligned} \delta (P)_{h}=d_{rh}-d_{lh}=0.332-0.312=0.02 \, \text {cm}. \end{aligned}$$

(3)

Again, its vertical coordinates $d_rv$ and $d_lv$ are the same, so $\delta (P) _ v=0$. Further, the ocular torsion vanishes for each eye, so the torsional disparity remains zero after the vertical shift of $F_a$.

The projection ray of P onto $P_a$ in the visual plane is also parallel to the subjective vertical horopter in this posture. Again in Fig. 6, the projection of P along the subjective vertical direction into the visual plane, $P_a$, is on the disparity line of disparity value $4(0.005)=0.02$ cm. It agrees with the horizontal disparity value obtained from the differences between the projections in the coordinates of the right and left eye image planes. It demonstrates that under the true vertical shift of the fixation point from the ERP, the visual space tilts together with the tilt of the subjective horopter by the same amount.

The next secondary posture is produced by shifting $F_a(0, 99.56, 1.72)$ horizontally to F(40, 99.56, 1.72), which is shown in Fig. 7.

Fig. 7

Under the horizontal shift of the fixation point from the resting eyes posture, the iso-disparity lines in the visual plane transform into hyperbolas. The subjective vertical horopter is tilted from the true vertical direction by about $\kappa =6.5^{\circ }$. See the text for a detailed discussion.

Now, the iso-disparity lines in the image plane change into hyperbolas for this secondary posture. Further, the subjective vertical horopter is tilted top-away by about $\kappa =6.5^{\circ }$ in the plane $-x+0.5y=6.8$ containing the subjective vertical horopter and the true vertical line (dashed line). Using the same argumentation as before, we see that the line through P and $P_a$, where P is chosen such that the longitudinal spatial disparity value of $4(0.005)=0,02$ cm agrees with the retinal horizontal disparity value of $\delta (P)_h=0.02$ cm and is parallel to subjective vertical horopter. In contrast to the vertical secondary posture, the vertical and torsional disparities are not zero after the horizontal shift: $\delta (P)_v=0.017$ cm and $\delta _t=\tau _r-\tau _l=0.01^{\circ }$.

Disparity spatial coordinates for eyes tertiary postures

Figures 8 and 9 show the disparity ellipses and hyperbolas for the tertiary postures when the resting eyes fixation point $F_a$ is shifted to positions with both horizontal and vertical components.

Fig. 8

The iso-disparity curves are ellipses, and the vertical horopter is tilted $\kappa=7.7^{\circ }$ for the fixation F(12, 33.56, 5.72). Details are given in the text.

Fig. 9

The disparity curves are hyperbolas, and the vertical horopter is tilted $\kappa =3.05^{\circ }$ for the fixation $F(-20,159.56,21.72)$. Note that the hyperbolic branches that pass through the nodal points $N_r$ and $N_l$ are not part of disparity curves.

In Fig. 8, for the fixation at F(12, 33.56, 5.72), the horizontal disparity value is $\delta (P)_{h}=0.02$ cm, the vertical disparity value is $\delta (P)_{v}=0.026$ cm and the torsional disparity value is $\delta _t=\tau _r-\tau _l=-0.61^{\circ }$. Also, the subjective vertical horopter is tilted in the plane $1.3x+0.12y=20$ by $\kappa =7.7^{\circ }$ bottom-away relative to the true vertical.

In Fig. 9, for the fixation at $F(-20, 159.56, 21.72)$, the horizontal disparity value is $\delta (P)_{h}=0.01$ cm, the vertical disparity value is $\delta (P)_{v}=-0.004$ cm and the torsional disparity value, $\delta _t=-0.11^{\circ }$. Finally, the subjective vertical horopter is tilted top-away in the plane $4.8x+2.3y=265$ by $\kappa =3.05^{\circ }$.

Simulation of ocular torsion

The actual geometric construction in GeoGebra’s simulation shown in Fig. 10 computes each of the two eyes’ ocular torsion for the case in Fig. 8 with the fixation point denoted here by $F^1$. By choosing the appropriate inputs of the fixation point, the ocular torsion of each eye was calculated for all simulations shown in Figs. 5, 6, 7, 8 and 9.

Fig. 10

The black frames in the ERP $(\textbf{i}_r,\textbf{j}_r,\textbf{k}_r)$ and $(\textbf{i}_l,\textbf{j}_l,\textbf{k}_l)$ for the right and left AEs are attached at the image planes optical centers $O_r$ and $O_l$. Their orientations agree with the upright head’s frame. The solid-colored frames are moving from the initial position of black frames for a given fixation $F^1$. The dashed-colored frames are obtained by rotating the solid-colored frames by moving $\textbf{b}$s vectors onto the $\textbf{j}$s vectors. The rotation axes are given by the vector $\textbf{n}^1_r$ and $\textbf{n}^1_l$, with the angle of rotations $\phi ^1_r$ and $\phi ^1_l$, are contained in the co-planar image planes of the ERP, parallel to the head frontal plane. The ocular torsion of each AE is $\tau ^1_r$ and $\tau ^1_l$.

In Fig. 10, the solid green frame for the right AE, $(\textbf{a}_r,\textbf{b}_r,\textbf{c}_r)$, and the brown frame for the left AE, $(\textbf{a}_l,\textbf{b}_l,\textbf{c}_l)$, are moving with the image planes from their position that initially in ERP agrees with the respective black frame for each AE.

To calculate each eye’s ocular torsion, the green and brown frames are rotated such that $\textbf{b}$s vectors are overlaid with $\textbf{j}$s vectors. The rotations are in the planes spanned by $\textbf{b}$s and $\textbf{j}$s vectors such that rotation axes containing the vectors $\textbf{n}^1_r$ and $\textbf{n}^1_l$ are perpendicular to these planes. The rotations result in two ‘primed’ frames shown in dashed lines. The rotations angles, $\angle (\textbf{j}_r,\textbf{b}_r)=18.436^{\circ }$ and $\angle (\textbf{j}_l,\textbf{b}_l)=23.821^{\circ }$ are denoted by $\phi ^1_r$ and $\phi ^1_l$.

Because $\textbf{n}^1_r$ is perpendicular to $\textbf{j}_r$ and $\textbf{n}^1_l$ is perpendicular to $\textbf{j}_l$; they are in the coplanar image planes for the ERP. The ocular torsion, defined by the angles $\angle \mathbf {(k}_r,\mathbf {c'}_r)=1.517^{\circ }$ and $\angle \mathbf {(k}_l,\mathbf {c'}_l)=2.133^{\circ }$, are denoted by $\tau ^1_r$ and $\tau ^1_l$ such that the torsional disparity $\delta ^1_t=\tau ^1_r-\tau ^1_l \approx -0.62^{\circ }$. All counterclockwise right-hand rotations are positive.

The results establish the geometrical calculations of the eyes’ ocular torsions for any binocular fixation executed from the ERP. The binocular ERP provides a neurophysiologically motivated new definition discussed in Section “Prior studies with the AE” for the eye’s primary position, originally part of Listing’s Law that controls ocular torsion but has been considered for the single eye.

Discussion of the simulated eyes postures

The simulation in Fig. 5 shows that the spatial disparity in the ERP is organized by true vertical planes passing through the frontal disparity lines. For any point in the binocular field, the vertical disparity is zero, and the horizontal disparity value is given by the frontal disparity line above which the point is located on the vertical plane. The backward tilt of the vertical horopter measured in³¹ for two subjects were about $6^{\circ }$ and $8.5^{\circ }$ for the observation distance of 1 m and $12^{\circ }$ and $16.5^{\circ }$ for the observation distance of 3 m. Figures 5 and 11 show a good agreement with Amigo results. Simulated here results also agree with the experimental results in⁴⁴. Thus, my results do not agree with Helmholtz and others^45,46 regarding the magnitude of the vertical horopter tilt.

Fig. 11

The iso-disparity curves and the vertical horopter simulated for the fixation F(0, 300.56, 1.72).

For the true vertical shift of the fixation point from the ERP shown in Fig. 6, the objective binocular space (and vertical planes above disparity lines) tilts by the same amount as the vertical horopter tilts at the fixation point. This visual space is seen subjectively as the same before the fixation point was shifted. In particular, the vertical and torsional disparities remain zero.

These results indicate that the curvature of visual space vanishes for those eyes’ postures, extending the 2D results in¹⁴ for the horizontal visual plane to the 3D results for these two postures. Moreover, they clearly show that the vertical disparity is not a binocular cue, agreeing with the general evidence from the random-dot-stereograms¹¹ and Wheatstone stereoscope¹⁰.

The situation is more complicated for the simulating horizontal shifts from the ERP shown in Fig. 7 for the secondary position with the eyes fixating on F(40, 99.56, 1.72). Although the values of the vertical and torsional disparities are small and the straight line at the fixation well approximates the hyperbola, it can affect our impression of being immersed in the ambient environment.

For the eyes shifted horizontally or vertically when both are in secondary positions, their disparity curves are markedly different after each shift, as shown in Figs. 6 and 7. This should be expected because the stereopsis is mainly related to the eyes’ lateral offset. My observation is that I use up-and-down eye movement more often without the head movement than when I rotate my eyes to the left or right. Is this behavioral dissimilarity connected to preserving the straight frontal disparity lines after vertical shifts and transforming them into hyperbolas after horizontal shifts? This question is worth investigating.

The one common feature of the vertical horopter orientations in Figs. 7, 8 and 9 is that under the change in fixation point, the vertical horopter’s tilt is specified in the computed vertical plane in each simulation. This aspect was never considered in experimental observations.

In the tertiary posture simulations shown in Figs. 6 and 9, the geometry of visual space is much more complicated, even under a simplifying assumption of a straight-line vertical horopter. However, in⁴⁷, the numerical analysis produced convex vertical horopters. The authors concluded that the vertical horopter is adaptive for perceiving convex, slanted surfaces at short distances. Nevertheless, the numerical methodology employed to discuss the vertical horopter curvature uses the Vieth-Müller circle (VMC) and the Hering-Hillebrand deviation coefficient H introduced in⁴⁸. However, each VMC and Ogles’ models of empirical horopter pass through the eye’s rotation centers rather than the nodal points, see^12,49.

One way to resolve the issue of the vertical horopter shape could be an experimental study similar to³¹ or⁴⁴ but with tested eye’ misalignment of the optical components and supported by the numerical methodology based on the AE model. Nevertheless, the results add to other contentions about the relevance of Luneburg’s assertion of the constant hyperbolic curvature visual space^50,51.

Discussion of Listing’s law in the binocular system with the AEs

Ocular torsion and Listing’s law

Historically, the torsion of the eye was determined by Donders, Listing, and Helmholtz between 1848 and 1867. Donders’s law is a general statement for a single eye with the head erect and looking at infinity; any gaze direction has a unique torsional angle, regardless of the path the eye follows to get there. Listing’s law, as formulated by Helmholtz in 1867, restricts Donders’s law by stating that visual directions of a single eyesight are related to the eye rotations so that from a reference position, all rotation axes lie in a plane, today referred to as the displacement plane. When the reference position is perpendicular to the displacement plane, the plane is referred to as the Listing plane, and the unique reference direction is the primary direction. The displacement plane for a given eye’s reference position was studied in the quaternion framework in^35,36 and reviewed in³⁷. It was only recently numerically simulated in³⁸.

Listing’s law has been extended in a few ways. In the first extension, Listing’s plane must tilt by half the angle of eye eccentricity from the primary direction^39,40. This monocular rule is known as the half-angle rule. The next two extensions involve the two eyes. First, Listing’s law is extended to “binocular” conjugate eye rotations with parallel gazes. Second, for the eyes’ posture with converging gazes on a near fixation point, each eye’s displacement plane turns by the amount proportional to the initial value of vergence angle^41,42. The extensions are known as binocular Listing’s law (L2), whereas the original law is L1.

However, the proportionality coefficients are controversial⁴¹. Further, the primary position, which is still a part of L2 law, corresponds to the eyes’ movements with parallel gazes, and, consequently, its neurophysiological significance has remained elusive despite its theoretical importance in the oculomotor research⁴³.

Listing’s law and the AE: Rodrigues’ vector computational framework

Foremostly, the AEs’ image planes for the ERP binocular posture are coplanar and constitute the frontal plane of the stationary upright head. The gaze direction of each eye in this posture passes through the point $F_{a}(0, 99.56, 1.72)$, expressed in centimeters, and is not perpendicular to this plane, so it is not Listing’s plane. It is not a displacement plane because its orientation is uniquely given by the AE’s parameters $\alpha$, $\beta$, $\gamma$, and $\varepsilon$.

The geometric theory developed here for binocular fixations is based on Euler’s rotation theorem. This theorem states that any two orientations of a rigid body with one of its points fixed differ by a rotation about an axis specified by a unit vector passing through the fixed point. I recall that fixed-axis rotations are curves of the shortest length (geodesics) on the manifold of the rotation (Lie) group SO(3) and, hence, are optimal; see³⁸ for a discussion of SO(3) geodesics in the context of eye rotations.

Euler’s rotation theorem is used here in the framework of Rodrigues’ vector for each of the two eyes constrained by their bifoveal fixations because it can be directly applied in simulations to rotate geometric objects in GeoGebra’s dynamic geometry environment. To introduce this framework, I start with the conclusion from Euler’s rotation theorem: Any rotation matrix R can be parametrized as $R(\phi ,\textbf{n})$ for a rotation angle $\phi$ around the axis $\textbf{n}$. This parametrization is unique if the orientation of $0<\phi <180^{\circ }$ is fixed. Usually, a counterclockwise (or right-hand) orientation is chosen to get angles’ positive values.

The pair $\rho =\cos (\phi /2), \textbf{e}=\sin (\phi /2)\textbf{n}$ is known as Euler-Rodrigues parameters and

$$\begin{aligned} \textbf{r}=\textbf{e}/\rho =\tan (\phi /2)\textbf{n} \end{aligned}$$

(4)

as Rodrigues’ vector, usually referred to as rotation vector⁵². Rodrigues proved that under composition $R=R_2R_1$, the corresponding Euler-Rodrigues parameters transform as follows⁵³,

$$\begin{aligned} \rho= & \rho _1\rho _2-\textbf{e}_1 \cdot \textbf{e}_2 \end{aligned}$$

(5)

$$\begin{aligned} \textbf{e}= & \rho _1\textbf{e}_2+\rho _2\textbf{e}_1+\textbf{e}_2\times \textbf{e}_1 \end{aligned}$$

(6)

Then, it is easy to see that the corresponding composition for Rodrigues’ vectors is

$$\begin{aligned} \textbf{r}=\textbf{r}_2 \circ \textbf{r}_1 =\frac{\textbf{r}_2+\textbf{r}_1+\textbf{r}_2 \times \textbf{r}_1}{1-\textbf{r}_1 \cdot \textbf{r}_2}. \end{aligned}$$

(7)

Further, $\textbf{r}^{-1}=-\textbf{r}$ and $\tan (-\phi /2)\textbf{n}=\tan (\phi /2)(-\textbf{n})$ define the same rotation vector.

Note that $(\cos (\phi /2),\sin (\phi /2)\textbf{n})$ is a unit quaternion that describes the rotation by $\phi$ around $\textbf{n}$, and Rodrigues used in 1840 the multiplication of quaternions to obtain (5) and (6). Remarkably, the quaternions were formally defined in 1843 by Hamilton, and vectors appeared late in the 19th century when Gibbs and Heaviside independently developed vector analysis.

The rotation vectors in the simulation shown in Fig. 10 for fixation $F^1(12,33.56,5.72)$ are

$$\begin{aligned} \textbf{r}^1_r= & \tan \left( \frac{\phi ^1_r}{2}\right) \textbf{n}^1_r \nonumber \\= & \tan \left( \frac{18.436^{\circ }}{2}\right) (0.461,0,-0.888)^t \nonumber \\= & (0.0748,0,-0.144)^t \end{aligned}$$

(8)

$$\begin{aligned} \textbf{r}^1_l= & \tan \left( \frac{\phi ^1_l}{2}\right) \textbf{n}^1_l \nonumber \\= & \tan \left( \frac{23.821^{\circ }}{2}\right) (0.338,0,-0.942)^t \nonumber \\= & (0.0712,0,-0.1986)^t. \end{aligned}$$

(9)

for the right and left eye, respectively.

Now, I can compute directly the torsion changes between tertiary eyes’ binocular fixations, which usually are obtained in oculomotor research using Listing’s L2.

To this end, I first show in Fig. 12 the simulation for the change of the fixation points from the resting eyes’ posture at $F_a(0, 99.56, 1.72)$ to $F^2(7, 53.56, 15.72)$.

Fig. 12

For the fixation $F^2(7,53.56,15.72$, the frames’ angles of rotations around the axes $\textbf{n}^2_r$ and $\textbf{n}^2_l$ are $\phi ^2_r=16.392^{\circ }$ and $\phi ^2_l=17.5^{\circ }$, respectively for the right and left AE. Similarly, the ocular torsions are $\tau ^{2}_{r}=0.86^{\circ }$ and $\tau ^{2}_{l}=1.33^{\circ }$.

Following similar calculations for $F^2$ as before for the fixation $F^1$, from results presented in Fig. 12, I obtain,

$$\begin{aligned} \textbf{r}^2_r= & \tan \left( \frac{\phi ^2_r}{2}\right) \textbf{n}^2_r \nonumber \\= & \tan \left( \frac{16.392^{\circ }}{2}\right) (0.866,0,-0.5)^t \nonumber \\= & (0.135,0,-0.0504)^t \end{aligned}$$

(10)

$$\begin{aligned} \textbf{r}^2_l= & \tan \left( \frac{\phi ^2_l}{2}\right) \textbf{n}^2_l \nonumber \\= & \tan \left( \frac{17.5^{\circ }}{2}\right) (0.866,0,-0.5)^t \nonumber \\= & (0.1332,0,-0.0769)^t. \end{aligned}$$

(11)

Then, frames rotations that align vector $\textbf{b}_r$ for $F^1(12,33.56,5.72)$ with with vector $\textbf{b}_r$ for $F^2(7, 53.56, 15.72)$ and similarly for vectors $\textbf{b}_l$s are calculated as follows. Let these Rodrigues’ vectors be denoted by $\textbf{r}^{12}_r$ and $\textbf{r}^{12}_l$ for the right and left eye, respectively. Thus, using the properties of Rodrigues’ vectors discussed above, I can first write $\textbf{r}^{12} \circ \textbf{r}^1 = \textbf{r}^2$ so that $\textbf{r}^{12} = \textbf{r}^2 \circ -\textbf{r}^1$ and then, using (7), I obtain

$$\begin{aligned} \textbf{r}^{12}_r= & \frac{\textbf{r}^2_r-\textbf{r}^1_r-\textbf{r}^2_r \times \textbf{r}^1_r}{1+\textbf{r}^2_r \cdot \textbf{r}^1_r} \nonumber \\= & (0.0589,-0.0154,-0.0922)^t \end{aligned}$$

(12)

$$\begin{aligned} \textbf{r}^{12}_l= & \frac{\textbf{r}^2_l-\textbf{r}^1_l-\textbf{r}^2_l \times \textbf{r}^1_l}{1+\textbf{r}^2_l \cdot \textbf{r}^1_l} \nonumber \\= & (0.0605,-0.0205,-0.0119)^t, \end{aligned}$$

(13)

so that from (4), (12) and (13), I have

$$\begin{aligned} \tan \frac{\phi _r^{12}}{2}= & \Vert \textbf{r}^{12}_r\Vert =0.1105 \end{aligned}$$

(14)

$$\begin{aligned} \tan \frac{\phi _l^{12}}{2}= & \Vert \textbf{r}^{12}_l\Vert =0.1349, \end{aligned}$$

(15)

which gives the corresponding angle values $\phi ^{12}_r=12.61^{\circ }$ and $\phi ^{12}_l=15.36^{\circ }$ for the right and left eye, respectively.

Finally, these Rodrigues’ vectors can be written as

$$\begin{aligned} \textbf{r}^{12}_r= & \tan \left( \frac{12.61^{\circ }}{2}\right) (0.533,-0.1394,0.8344)^t \end{aligned}$$

(16)

$$\begin{aligned} \textbf{r}^{12}_l= & \tan \left( \frac{15.36^{\circ }}{2}\right) (0.449,-0.1518,-0.8807)^t. \end{aligned}$$

(17)

and the ocular torsions when the eyes fixation changes from $F^1$ to $F^2$ can be calculated as $\tau ^{12}_r =\tau ^{1}_r-\tau ^{2}_r=0.66^{\circ }$ and $\tau ^{12}_l =\tau ^{1}_l-\tau ^{2}_l=0.80^{\circ }$ using the values given in Figs. 10 and 12.

Fig. 13

The half-angle rule for the fixation changing from $F^1(12, 33.56, 5.72)$ to $F^2(7, 53.56, 15.72)$. Rodrigues’ vectors $\textbf{r}^{12}_r$ and $\textbf{r}^{12}_l$ rotate visual axes for $F^1$ to align them with the visual axes for $F^2$. Green dots $\textbf{r}^{12}_rF^1$ and $\textbf{r}^{12}_lF^1$ are rotations of $F^1$ by the indicated Rodrigues’ vectors. The angles $\kappa _r =8.1^{\circ }$ and $\kappa _l=8.7^{\circ }$ approximate the half-angle of $\xi _r=15.8^{\circ }$ and $\xi _l=18.9^{\circ }$ for the right and the left eye, respectively. The half-angle rule—the rotation of Rodrigue’s vectors $\textbf{r}_{r}^{12}$ and $\textbf{r}_{l}^{12}$ out of the ERP plane—which is approximate because of the eyes’ misaligned optical components, is demonstrated by the angles $\pi _r$ and $\pi _l$ (angles between bisectors and Rodrigues’ vectors) that are very close to right angles and the perpendicularity of $l_r$ and $m_r$ as well as $l_l$ and $m_l$ lines. The construction of the angles approximating the half-angle rule is explained in the text.

I conclude this section with the half-angle rule. It describes the eye rotation between tertiary configurations, which Helmholtz first discussed in connection to Listing’s law. It was attributed to the non-commutativity of 3D rotations, see⁵⁴. The half-angle rule was first proved in⁵⁵ using quaternion representation of rotations. Also, its alternative proof was given by more general nonlinear dynamics methods along with an extensive discussion in³⁸. These proofs were only provided for one eye. Here, the half-angle rule is demonstrated in GeoGebra simulations for binocular fixations using Euler’s rotation theorem implemented in Rodrigues’ vector framework. It is shown in Fig. 13.

To discuss the half-angle approximations for both eyes, the eccentricity of the fixation point $F^1$ must be resolved in the ERP such that all angles for each eye are located in the same plane. I recall that in ERP posture, the image planes are coplanar and define the frontal plane of the binocular system. In Fig. 13, the gray plane contains Rodrigues’ vector $\textbf{r}_{r}^{12}$ and the line $l_r$ perpendicular to the image plane at the optical center $O_r$. Similarly, the brown plane contains Rodrigues’ vector $\textbf{r}_{l}^{12}$ and the line $l_l$ perpendicular to the image plane at the optical center $O_l$. Then, for each eye, the visual axes of $F^1$ are rotated about the intersection line of the frontal plane and gray plane for the right eye and the brown plane for the left eye—the lines $m_r$ and $m_l$ respectively—such that they align with the appropriate gray or brown plane. These rotated visual axes give rays through $F_r^1$ for the right eye and through $F_l^1$ for the left eye. Then, between $l_r$ and $l_l$ and the corresponding rays, the bisecting lines $b_r$ and $b_l$ are obtained. The half-angle rule in Fig. 13 is demonstrated by the angles $\pi _r$ and $\pi _l$ (angles between bisectors and Rodrigues’ vectors) that are very close to right angles and the perpendicularity of $l_r$ and $m_r$ as well as $l_l$ and $m_l$ lines.

This half-angle rule is a corollary of Euler’s rotation theorem for the ERP posture, which is an appropriate substitution for the primary or displacement planes underlying Listing’s law. The real significance of this rule is the fact that it is binocular: it is applied to each eye and transforms the bifoveal fixations from $F^{1}(12, 33.56, 5.72)$ to $F^{2}(7, 53.56, 15.72)$ by rotating with Rodrigues’ vectors (16) and (17). However, the small discrepancy mentioned in the caption of Fig. 13 results from the eye’s misaligned optical components. Importantly, the nonzero vergence coefficients that define the tilts of the initial Listing plane for each eye in the binocular extensions L2 that are controversial are not needed in the approach used here.

Conclusions

In healthy human eyes, the fovea is not located on, and the lens is tilted away from the optical axis. It is well known that the lens’ horizontal tilt partially compensates for the optical aberrations resulting from the fovea anatomic displacement from the posterior pole. The recent study in¹⁴ established the geometric theory of horizontal retinal correspondence’s spatial coordinates (iso-disparity curves) in the binocular system with the asymmetric eye (AE) model that comprises horizontal misalignment of optical components. That study revealed that the distribution of the iso-disparity lines is invariant to the AE horizontal misalignment of the fovea and the lens. It demonstrated for the first time that the observed asymmetry of the retinal correspondence in human eyes is caused by misaligned fovea and lens.

The study presented here extended this theory by including the vertical misalignment of the eye’s optical components, providing the comprehensive geometric construction of 3D spatial coordinates of the retinal disparity correspondence and elucidating their functional role in binocular perception. In particular, the vertical misalignment of optical components determines the orientation of the visual plane and, more fundamentally, the 3D lens’ tilt replaces classic Helmholtz shear theory that explains the inclination of the perceived vertical horopter top-away from the true vertical but is less precise due to the global asymmetry of the eyeball resulting from the displaced fovea from the posterior pole by $5^{\circ }$. The criterion of the retinal vertical correspondence provided by the 3D lens’ tilt and the resulting inclination of the perceived vertical horopter agrees with experimental results in^31,44. Thus, it does not agree with Helmholtz and others^45,46 regarding the magnitude of the vertical horopter tilt. Further, the simulation results indicate that the perceived vertical should be identified with the apparent vertical horopter, which disagrees with the opposite statement in⁴⁶.

Ocular torsion for the AE binocular posture is geometrically defined using Euler’s rotation theorem, which closely approximates the len’s rolling eye movement and is simulated in GeoGebra in the framework of Rodrigues’ vector. It provides a consistent binocular formulation of Listing’s law for the first time. In particular, it replaces the Listing plane, defined ad hoc for a single eye and then extended to “binocular” eye movement with parallel gazes, with the eye’s resting posture or ERP. The ERP is the binocular posture that, for the average asymmetry angles of the AE, numerically corresponds to the eyes’ resting vergence posture, in which the eye muscles’ natural tonus resting position serves as a zero-reference level for convergence effort¹⁶. The binocular ERP provides a new definition for Listing’s law eye’s primary position—gaze line perpendicular to the Listing plane containing rotation vectors of eye movement executed from the primary position. Note that the eye’s primary position neurophysiological significance has remained elusive despite its theoretical importance in the oculomotor control⁴³.