The distribution of empirical corresponding points in the two retinas has been well studied along the horizontal and the vertical meridians, but not in other parts of the visual field. Using an apparent-motion paradigm, we measured the positions of those points across the central portion of the visual field. We found that the Hering–Hillebrand deviation (a deviation from the Vieth–Müller circle) and the Helmholtz shear of horizontal disparity (backward slant of the vertical horopter) exist throughout the visual field. We also found no evidence for non-zero vertical disparities in empirical corresponding points. We used the data to find the combination of points in space and binocular eye position that minimizes the disparity between stimulated points on the retinas and the empirical corresponding points. The optimum surface is a top-back slanted surface at medium to far distance depending on the observer. The line in the middle of the surface extending away from the observer comes very close to lying in the plane of the ground as the observer fixates various positions in the ground, a speculation Helmholtz made that has since been misunderstood.

*n*points in an image there are

*n*

^{2}possible matches, which becomes a very large number with complex stimuli such as random-element stereograms (Julesz, 1971). One part of the solution to this problem is the existence of points in the two retinas with a special physiological relationship: For each point in one retina, there is a point in the other that when stimulated gives rise to the same perceived direction (i.e., the points appear superimposed in visual space). These pairs, which are called

*corresponding points,*have special status for binocular vision: (1) matching solutions between the two eyes' images are biased toward them (Brewster, 1844; Prince & Eagle, 2000); (2) the region of single vision straddles them (Fischer, 1924); and (3) the precision of depth estimates from disparity is highest for locations in space that project to those points (Badcock & Schor, 1985; Blakemore, 1970; Breitmeyer, Julesz, & Kropfl, 1975; Ogle, 1953; Schumer & Julesz, 1984; Westheimer, 1982). We measured the locations of corresponding points across the central visual field and used the measurements to estimate the shape of the surface for which human stereopsis is best suited. Such information provides important insight into how stereopsis functions in the natural environment. It can also be used to design workstations that maximize visual performance and reduce viewer fatigue (Ankrum, Hansen, & Nemeth, 1995).

*horopter*(Helmholtz, 1925; Howard & Rogers, 2002; Tyler, 1991). The geometric horopter is constructed by projecting rays out of the eyes from pairs of geometric corresponding points and finding the intersections of those rays; only a subset of rays yields intersections and the positions of the intersections obviously depend on eye position (Figure 1). The empirical horopter is constructed by projecting rays from empirical corresponding points; again only a subset yields intersections and again the intersection positions depend on eye position.

*Hering–Hillebrand deviation*(e.g., Ames, Ogle, & Gliddon, 1932; Ogle, 1932; Shipley & Rawlings, 1970; see Equation 1). Because the retinal positions of empirical corresponding points do not shift with eye movements (Hillis & Banks, 2001), the Hering–Hillebrand deviation remains constant when expressed in angular units. However, the curvature of the empirical horizontal horopter changes with vergence eye movements because of the relationship between disparity and distance (Howard & Rogers, 2002; Ogle, 1950). With decreasing vergence (increasing distance), it becomes less concave and eventually becomes convex. Relative to geometric corresponding points, empirical corresponding points near the vertical meridians of the retinas are sheared horizontally such that uncrossed disparities are required to stimulate corresponding points above the fovea and crossed disparities below, and the magnitude of these disparities increases linearly with elevation. This vertical gradient of horizontal disparity has been called the

*Helmholtz shear*. It causes a top-back slant of the vertical horopter (Helmholtz, 1925; Nakayama, 1977; Siderov, Harwerth, & Bedell, 1999; Tyler, 1991). Because of the relationship between disparity and distance, the slant of the empirical vertical horopter increases with fixation distance.

*optimum surface*because it is the surface for which the precision of stereopsis should be highest.

*Ae*

^{2}+

*Be,*where

*e*is eccentricity along the meridian and

*A*and

*B*are constants. We did not have a constant term in the fitting equation in order to constrain the fits to have a zero offset at their midpoints and thereby eliminate possible discontinuities at the center where the meridians all meet. The polar-Cartesian conversion required interpolation for locations in the Cartesian grid where we did not actually collect data. The interpolated disparity was the weighted average of measured disparities at corresponding eccentricities on the two neighboring radial meridians. From those disparities in the grid, we obtained estimates of the Hering–Hillebrand deviation, Helmholtz shear, and vertical-disparity shear. We represented the vertical-disparity pattern in the form of a horizontal gradient because the vertical-disparity pattern of a plane viewed with symmetrical vergence is well summarized that way. The results are shown in Figures 6, 7, and 8. To estimate the error on the parameter fits, we used a bootstrapping method in which we randomly removed two points from each meridional data set before performing the fits described above. With the remaining five points (including the fixation point) on each meridian, we ran the parameter-fitting routine. We did this 50 times deriving an estimate of the Hering–Hillebrand deviation as a function of elevation, Helmholtz shear as a function of azimuth, and vertical-disparity shear as a function of elevation each time. We then used those 50 estimates to estimate the standard deviations of the fitted parameters.

*H,*based on relative horizontal disparities in the left and the right eyes,

*α*

_{L}and

*α*

_{R}, and the effective image magnification in the right eye relative to the left,

*R*:

*D*as the difference between

*α*

_{L}and

*α*

_{R}minus an elevation-specific disparity offset

*D*

_{0}that we used to offset the effect of the Helmholtz shear:

*D,*

*H,*and

*R*that provided the best least-squares fit to the data. When

*R*= 1 (equal magnification in the two eyes) and

*H*=

*D*= 0, Equation 3 describes the situation in which empirical and geometric corresponding points are the same; the empirical horopter for the horizontal meridian then becomes the Vieth–Müller circle. For

*H*> 0, the horopter is less concave. We needed the parameter

*D*

_{0}because the equation for the Hering–Hillebrand deviation assumes that the disparity is zero at an azimuth of zero (i.e., that

*D*= 0 for

*α*= 0). Because of the Helmholtz shear, this assumption is false for all non-zero elevations.

*R*did not vary with elevation, but that it was very slightly greater than 1 for all three subjects. The slight deviation from 1 could have been caused by a very small difference in the distances of the left and right CRTs to their respective eyes.

*H*values are plotted in Figure 6. At an elevation of zero (i.e., on the horizontal meridian),

*H*is +0.25 for observer PRM, +0.36 for KMS, and −0.11 for HRF.

*H*is generally +0.10 to +0.30 in observers with normal binocular vision (Ogle, 1950), so the data from two of our observers are consistent with previous measurements. Of greatest interest, however, is the fact that

*H*does not vary systematically with elevation. This result is consistent with the conclusions of Grove et al. (2001) and Ledgeway and Rogers (1999).

Horizontal vergence (deg) | Vertical vergence (deg) | Torsional vergence (deg) | Distance (cm) | |
---|---|---|---|---|

PRM | 2.40 | 0.01 | −0.20 | 155 |

KMS | 1.33 | 0.06 | −0.53 | 279 |

HRF | 4.15 | 0.00 | −0.51 | 90 |

*H,*the Helmholtz shear, and vertical-disparity shear for two of their three observers (the third had too few measured points). The upper row of Figure 9 plots the best-fitting

*H*as a function of elevation. Their

*H*values were similar to ours exhibiting a tendency toward positive values and no systematic change with elevation. The middle row of Figure 9 plots the best-fitting Helmholtz shear as a function of azimuth. They observed the same effect we did: larger shear disparities at 0° than at other azimuths. The lower row plots vertical-disparity shear as a function of elevation. There was a small variation in shear with elevation: smaller values at zero elevation than at others. That pattern is not adaptive for any viewing distance (see Figure 4D). The vertical-disparity shear they observed was quite small, so their observations were similar to ours (Figure 8): they showed no substantial variation of vertical disparity with elevation.

*θ*/2 in the left eye and −

*θ*/2 in the right eye, so the left-right difference is

*θ*. When

*θ*

*is*positive, as it usually is, we will refer to this correspondence shift as extorsion. The projections of the extorted retinal meridians through the centers of the eyes are extorted planes. The two planes intersect in a line beneath the observer when

*θ*is positive and in a line above when

*θ*is negative. That line of intersection is the empirical vertical horopter. With the eyes gazing parallel to the ground at an infinite distance, the vertical horopter is parallel to the ground and its elevation relative to the eyes is

*i*is the interocular distance. This result was described by Cooper and Pettigrew (1979, their Equation 1), Helmholtz (1925), Howard and Rogers (2002, chap. 15, their Equation 9), and others. If the elevation corresponds to the observer's eye height, the empirical horopter lies in the ground with the eyes in parallel gaze. This arrangement would be advantageous because it causes the region of single vision and best stereopsis to coincide with the predominant environmental surface. The value of

*θ*that places the horopter in the ground is

*h*is eye height. For heights and interocular distances that are common for humans (

*h*= 160 cm;

*i*= 6.5 cm),

*θ*

_{crit}is 2.32°, a value that is reasonably consistent with empirical observation (Grove et al., 2001; Helmholtz, 1925; Nakayama, 1977; our data in Figure 7). Helmholtz described this result:

“In one single instance the horopter is a surface, which in fact is a plane; and that is when the point of fixation is in the median plane and at an infinite distance, and the retinal horizons, as is usually the case or practically so at any rate if the eyes are normal, are both in the visual plane. Then this horopter-plane will be parallel to the visual plane, its distance from the latter depending on the amount of divergence of the apparently vertical meridians of the visual globes of the two eyes; that is, it will contain the line of intersection of these two median planes and will usually be practically the same as the horizontal plane on which the observer is standing, provided his eyes are normal and directed straight toward the horizon” (p. 424).

*θ*is constant and equal to

*θ*

_{crit}, the slant of the empirical vertical horopter varies with fixation distance such that it always runs through a point at the observer's feet. This claim, which has been attributed to Helmholtz, is not exactly correct. To see this, reconsider Figure 10. As the eyes converge while looking parallel to the ground, they rotate about vertical axes. The extorted planes contain the fixation axes, so the planes rotate with the eyes (not shown in the figure). As a consequence, the angle between the planes changes; it is no longer

*θ*(which is a retinal entity). The elevation of the point on the horopter directly under the eyes becomes:

*μ*is the horizontal vergence, the angle between the lines of sight. (To see this, note that the planes rotate inward when the eyes converge. The point on the horopter directly under the eyes was rotated there from a position at coordinates (sin(

*μ*/2)(

*i*/2), cos(

*μ*/2)(

*i*/2)) under parallel gaze. The elevation of points on the black planes in Figure 10 changes linearly from −

*i*/2tan(

*θ*/2) to 0 over a horizontal distance of

*i*/2. Equation 6 follows directly. Thus, for a fixed

*θ,*the elevation at which the vertical horopter crosses under the observer depends on fixation distance, an observation that is inconsistent with several figures in the literature (e.g., Fig. 10 in Cooper & Pettigrew, 1979; Fig. 15.29 in Howard & Rogers, 2002; Fig. 5 in Nakayama, 1977; Fig. 16.9 in Nakayama, 1983). Figure 11 shows the variation of

*e*with fixation distance when

*θ*is constant at

*θ*

_{crit}. The differences between elevations predicted by Equations 4 and 6 are very small until the fixation distance becomes less than 50 cm, so the error in previous work has little practical significance.

“Another matter that must be mentioned here is that, when a person holds his body and head erect and looks at a point on the floor-plane which is also in the median plane of the head, the entire floor-plane is not the horopter in this case, but yet the entire rectilinear part of the horopter does lie in this plane” (p. 425).

*θ*that places the empirical vertical horopter in the ground plane. He was then considering the eye movements that would take fixation from infinity to near points in the ground and mid-sagittal plane; these movements contain a horizontal component (to converge to a shorter distance) and a vertical component (to place fixation low enough to hit the ground). There is an eye movement that would not alter the head-centric position and orientation of the extorted planes projected from the sheared vertical meridians and therefore would keep the horopter in the same head-centric location: The rotation axis has to be perpendicular to the tilted meridian in each eye when the eyes are in primary position. By rotating about that particular axis, an eye's line of sight remains in the extorted plane; the eyes rotate through short angles to look at distant ground points and through large angles to look at near points. Because the extorted planes do not move relative to the head, their intersection—the empirical vertical horopter—remains precisely in the ground. Notice that the required axis is parallel to the head's coronal plane, which means that the movement follows Listing's law ( Figure 10). Although he was not explicit in the above quotation, we think Helmholtz was referring to Listing's movements when “a person … looks at a point on the floor-plane” because he described such movements as normal eye movements throughout this section of the book. The eye movements considered explicitly by Cooper and Pettigrew (1979) and implicitly by other modern authors were only horizontal and did not place fixation in the ground.

*θ*) and the type of binocular eye movement affect the orientation of the vertical horopter with respect to the ground. The insets in Figure 12A illustrate the viewing situation. A standing viewer with the head upright looks down to place fixation in the ground at some distance. The slant of the empirical vertical horopter (short red line segments) depends on the value of

*θ*and the distance to the fixation point. The horopter is of course in the mid-sagittal plane. The angle between the empirical horopter and the ground is

*ρ,*so when

*ρ*is zero, the horopter lies in the ground. The upper two panels of Figure 12 show the relationship between

*θ*and

*ρ*when eye position follows Listing's law. The horopter lies precisely in the ground plane when

*θ*= 2.32° and lies close to the ground when

*θ*is between 1.5° and 6°. The value of

*θ*measured in different individuals nearly always falls within that range (Grove et al., 2001; Ledgeway & Rogers, 1999; Nakayama, 1977; our Figure 7). The lower left panel of Figure 12 shows the relationship between

*θ*and

*ρ*when eye position does not follow Listing's law. There are many such eye positions, but we show those with both eyes' Helmholtz torsion being zero. In this case, the horopter does not lie precisely in the ground plane for any value of

*θ,*but it comes closest at a value of 2.32° and is again approximately in the ground for values between 1.5° and 6° at all but the nearest distances. Thus, eye positions that do not obey Listing's law can still keep the horopter in the ground at all but near distances.

*θ*and

*ρ*when the cat fixates in the plane of the ground with eye positions following Listing's law. The horopter lies exactly in the ground when

*θ*is 10.9° (Equation 5) and is close for values between 5° and 20°.

*θ*

_{crit}) keeps the vertical horopter in the ground as an upright viewer makes Listing's eye movements to look from one mid-sagittal location in the ground to another. This is clearly advantageous because it places the region of single vision and best stereopsis in the predominant environmental surface. Interestingly, computer-vision algorithms have adopted a similar shear for two-camera systems used to guide vehicles across the ground (Koller, Luong, & Malik, 1994).

*H*that describes the curvature of the horopter relative to the Vieth–Müller circle.

*H*is 0.10–0.15 on average (Ames et al., 1932; Fischer, 1924; Grove et al., 2001; Hillis & Banks, 2001; Ledgeway & Rogers, 1999; our Figure 7). To determine how well the horizontal extension of the horopter fits the ground plane, we calculated the minimum-disparity horopter using values of 2.3° for the Helmholtz shear and 0.10 for the Hering–Hillebrand deviation. For simplicity, we assumed that the shear was the same for all azimuths and the deviation the same for all elevations; there is of course some evidence that the former assumption is slightly incorrect (Figures 8 and 9).

*H,*the curvature of the horizontal extension of the horopter varies with distance. There is one distance—the abathic distance—at which the extension is planar. That distance is

*d*=

*i*/

*H,*where

*i*is interocular distance (Ogle, 1950). For distances less than the abathic distance, the extension is concave; for greater distances, it is convex. Thus, it is impossible for the horizontal extension of the horopter to be planar at all distances. Movie 2 illustrates this fact. In the left panel, a standing viewer is fixating in the ground plane and the mid-sagittal plane. The blue lines represent the two eyes' lines of sight and the green dashed line is an earth-horizontal line. The distance to the fixation point is greater than the abathic distance of 64 cm, and as a result, the horizontal extension of the horopter is convex such that it lies beneath the ground plane on the sides. The horopter surface in this case is a smooth ridge rather than a plane. The curvature of the ridge would increase with increasing fixation distance, so at long distances the horopter surface would be well below the ground on the sides. Thus, the horizontal extension of the horopter cannot in general be coincident with the plane of the ground.

*H*is constant as a function of elevation ( Figure 7 and Grove et al., 2001) and the Helmholtz shear is greater than zero (Figure 8 and many reports), the horopter has different curvatures at different elevations relative to the fixation point.

*H*values greater than the value of 0.1 that we assumed here (Ames et al., 1932; Hillis & Banks, 2001; Ogle, 1932; Shipley & Rawlings, 1970). In those cases, the horopter comes close to being coincident with a yet nearer surface than shown in the right panel of Movie 2.