2.1.

Fixation Disparity

FD is the difference between actual $\left({\alpha }_{\mathrm{A}}\right)$ and ideal $\left({\alpha }_{\mathrm{I}}\right)$ vergence angles [see Eq. 1]. The former is the vergence angle measured when the observer looks at the screen (lines of gaze may align on the screen plane or beyond or before it); the latter is the theoretical expected vergence angle with perfect alignment on the screen plane (i.e., zero fixation disparity):

The main parameters involved in the computation of ${\alpha }_{\mathrm{A}}$ and ${\alpha }_{\mathrm{I}}$ are sketched in Fig. 1 (where ${\alpha }_{\mathrm{A}}$ is represented in a case simulating eso-disparity).

Fig. 1

Elements (planes, points, segments, and angles) involved in the trigonometric computations. The ideal vergence angle $\left({\alpha }_{\mathrm{I}}\right)$ is sketched by the black dash-dotted line; the actual vergence angle $\left({\alpha }_{\mathrm{A}}\right)$ is presented in the case of hyperconvergence (eso-disparity) by a solid black line.

In Fig. 1, RE and LE are the right eye and left eye, respectively. The line joining the rotation centers ( ${\mathrm{C}}_{\mathrm{R}}$ and ${\mathrm{C}}_{\mathrm{L}}$ ) of the two eyes is the interocular baseline. The segment $PD$ is the distance between the eye rotation centers ${\mathrm{C}}_{\mathrm{R}}$ and ${\mathrm{C}}_{\mathrm{L}}$ ; since this cannot be measured directly, interpupillary distance was used. S is the PC screen plane, which is parallel to the interocular baseline; $b$ is the distance between the screen plane and the corneal vertex; $d_c$ is the distance between the corneal vertex and the eye rotation center; ² $d$ is the distance between plane S and the interocular baseline; it is calculated as the sum of $b$ and $d_c$ . M is a point midway between ${\mathrm{C}}_{\mathrm{R}}$ and ${\mathrm{C}}_{\mathrm{L}}$ . ${\mathrm{S}}_0$ is the orthogonal projection of M on plane S and corresponds to the center of the horizontal meridian of the PC screen. ${\mathrm{S}}_{\mathrm{R}}$ and ${\mathrm{S}}_{\mathrm{L}}$ represent projections on the screen of the RE and LE lines of gaze, respectively. ³ ${\mathrm{S}}_m$ represents the point of ideal vergence, where ${\mathrm{S}}_{\mathrm{R}}$ coincides with ${\mathrm{S}}_{\mathrm{L}}$ (see dashed lines in Fig. 1). If ${\mathrm{S}}_{\mathrm{R}}$ and ${\mathrm{S}}_{\mathrm{L}}$ do not coincide, the lines of gaze may either cross in front of plane S (as in Fig. 1) or behind it (not represented). In either case, ${\mathrm{S}}_m$ is the midpoint of segment ${\mathrm{S}}_{\mathrm{R}}{\mathrm{S}}_{\mathrm{L}}$ ; $y$ is the measure of this segment. When $y$ is equal to zero, the binocular alignment is perfect (i.e., the lines of gaze converge on ${\mathrm{S}}_m$ ); the greater $y$ , the greater the error in convergence. $j$ is the distance between ${\mathrm{S}}_0$ and ${\mathrm{S}}_m$ .

In the sketched case of eso-disparity, F is the actual fixation plane; ${\mathrm{F}}_{\mathrm{A}}$ is the actual fixation point; ${\mathrm{S}}_{\mathrm{A}}$ is the orthogonal projection of ${\mathrm{F}}_{\mathrm{A}}$ on plane S; and $k$ is the distance between ${\mathrm{S}}_0$ and ${\mathrm{S}}_{\mathrm{A}}$ .

Of the above parameters, $y$ , $PD$ , and $d$ can be collected empirically. The remaining parameters are calculated by applying trigonometric rules, as reported below.

2.2.

Computation of Ideal Vergence Angle $\left({\alpha }_{\mathrm{I}}\right)$

The ideal vergence angle $\left({\alpha }_{\mathrm{I}}\right)$ is the angle formed by the two lines of gaze when they coincide on the stimulus plane (i.e., fixation plane F coincides with screen plane S). In this case, ${\alpha }_{\mathrm{I}}$ is equal to ${\alpha }_{\mathrm{A}}$ , and FD is equal to zero.

Three cases of ${\alpha }_{\mathrm{I}}$ will be considered.

Fig. 2

Ideal vergence angle; three cases are depicted. In the first case $\left({\alpha }_{\mathrm{I}0}\right)$ , point ${\mathrm{F}}_{\mathrm{A}0}$ (as defined in Fig. 1) is on the midline between the eyes; its projection on plane $S$ coincides with ${\mathrm{S}}_0$ , so $j=0$ (note that $S_{\mathrm{I}0}$ in this figure corresponds with $S_0$ in Fig. 1). In the second case $\left({\alpha }_{\mathrm{I}1}\right)$ , segment $j$ is less than $PD∕2$ . In the third case $\left({\alpha }_{\mathrm{I}2}\right)$ , segment $j$ is greater than $PD∕2$ .

Equations 3, 4 are equivalent for computing the binocular vergence angle. They can also be applied when ${\mathrm{S}}_m$ coincides with ${\mathrm{S}}_0$ (in this case, $j$ is equal to zero), and also in the particular case of $j=PD∕2$ .

2.3.

Determination of Actual Vergence Angle $\left({\alpha }_{\mathrm{A}}\right)$ from Gaze Position Recorded by Infrared Eye-Trackers

The actual vergence angle $\left({\alpha }_{\mathrm{A}}\right)$ is determined by the LE and RE lines of gaze when the observer looks at the screen.

Infrared eye-trackers provide information on the projection of the line of gaze; gaze position is represented in bidimensional coordinates relative to the surface of a PC screen.

Again, we will consider three cases:

Fig. 3

Actual vergence angle ${\alpha }_{\mathrm{A}}$ . In this case, point ${\mathrm{F}}_{\mathrm{A}}$ lies on segment ${\mathrm{MS}}_0$ , the projection of point ${\mathrm{F}}_{\mathrm{A}}$ on plane $\mathrm{S}$ coincides with ${\mathrm{S}}_0$ , and $k=0$ . ${\mathrm{S}}_0$ in this figure corresponds with ${\mathrm{S}}_0$ in Fig. 1 and ${\mathrm{S}}_{\mathrm{I}0}$ in Fig. 2.

Fig. 4

Actual vergence angle; Two cases are depicted other than those for the condition presented in Fig. 3. In the first case $\left({\alpha }_{\mathrm{A}1}\right)$ , segment $k$ is less than $PD∕2$ . In the second case $\left({\alpha }_{\mathrm{A}2}\right)$ , segment $k$ is greater than $PD∕2$ .

Equations 10, 13 are equivalent and effective even for computing convergence angles when ${\mathrm{F}}_{\mathrm{A}}$ lies on segment ${\mathrm{MS}}_0$ , so Eq. 10 can be used as the final equation for the actual vergence angle in all cases.

2.4.

Final FD Equation

At the beginning of Sec. 2.1, FD was defined as the difference between the convergence angles ${\alpha }_{\mathrm{A}}$ and ${\alpha }_{\mathrm{I}}$ . Therefore, to compute FD, the final step is to substitute Eq. 1 with both Eq. 10 (computing ${\alpha }_{\mathrm{A}}$ ) and Eq. 3 (computing ${\alpha }_{\mathrm{I}}$ ):

In this final equation, the terms corresponding to an observer’s interpupillary distance and viewing distance ( $PD$ and $d$ , respectively) are known; the gaze positions ( ${\mathrm{S}}_{\mathrm{L}}$ and ${\mathrm{S}}_{\mathrm{R}}$ ) are empirically collected by eye movement recording. Then the value of $y$ is obtained as the difference between eye gaze positions (i.e., computed as ${\mathrm{S}}_{\mathrm{L}}-{\mathrm{S}}_{\mathrm{R}}$ ); $k$ is computed as $PD∕2-[{\mathrm{S}}_{\mathrm{L}}-({\mathrm{S}}_0-PD∕2)](d-x)∕d$ ; $x$ is solved with Eq. 8, i.e., $dy∕(PD+y)$ ; and $j$ is computed as the difference between ${\mathrm{S}}_0$ and ${\mathrm{S}}_m$ [where the value of ${\mathrm{S}}_m$ is computed as $({\mathrm{S}}_{\mathrm{L}}+{\mathrm{S}}_{\mathrm{R}})∕2$ ].

The appendix contains an example showing how to implement Eqs. 3, 10 to calculate FD in a commonly used datasheet program (Microsoft Excel).

3.1.

Measurement of FD during Target Fixation

This section presents an experiment in which FD was computed using Eq. 14. To obtain gaze position data ( ${\mathrm{S}}_{\mathrm{L}}$ and ${\mathrm{S}}_{\mathrm{R}}$ ), binocular eye movements were recorded in one participant as he performed a fixation task. Gaze position data ${\mathrm{S}}_{\mathrm{R}}$ and ${\mathrm{S}}_{\mathrm{L}}$ were used as known terms of the equation together with the participant’s interpupillary distance and the viewing distance. Care was taken to perform binocular calibration. A calibration matrix with dichoptic nonius lines was used to allow eye alignment. Calibration data were acquired only when the participant perceived the alignment of nonius lines. Then the fixation task was performed by looking at the target on the screen. To check for the goodness of the measurement, the task was repeated with a near target. To give a magnified example of lines of gaze crossing in front of the screen plane, the fixation target was displayed approximately halfway between the participant and the screen, thus producing a result that mimicked strong eso-disparity.

An actual FD approaching zero was expected when the participant looked at the far target on the screen. This expectation was due to the use of nonius lines and dichoptic alignment not only for calibration points, but also for the task fixation points. In fact, to appreciate equation efficacy, it was necessary to use a target that elicited an actual FD comparable to the ideal FD.

An eso-disparity of approximately $6\mathrm{deg}$ was expected during fixation of the near target. In fact, this target was located halfway between the screen plane and the eye plane. While ${\mathrm{S}}_{\mathrm{L}}$ and ${\mathrm{S}}_{\mathrm{R}}$ were referred to the PC screen, the lines of gaze crossed in front of the screen and produced a case of eso-disparity.

As shown below, mean FD while the participant looked at the screen tended toward zero, with the measured vergence angle nearly equal to the ideal vergence angle. On the contrary, while the participant looked at the near target, FD was magnified in the direction of eso-disparity, because the viewing distance reference in the equation was maintained as the eye-to-screen distance.

3.2.

Participant

One of the authors, F.Z, a healthy $39\text{-}\text{year}$ -old male, participated in the experiment. His eyes were healthy based on direct ophthalmoscopy and slit-lamp observation. The interpupillary distance was $60\mathrm{mm}$ , and the noncyclopegic autorefractometry measure (Retinomax 2; Nikon) was OD $\mathrm{sf}+0.50\mathrm{cil}\text{-}0.25\mathrm{ax}5°$ , OS $\mathrm{sf}+0.50\mathrm{cil}\text{-}0.50167°$ . His uncorrected monocular visual acuity was 20/18 (corresponding to −0.05 logMar; MAR=minimum angle of resolution), and his sighting eye (tested by the hole-in-the-card test, according to Griffin and Grisham⁸) was right. Based on von Graefe’s method of measurement, the values of far and near horizontal heterophoria were ${2.0}^{\Delta }$ exo and ${10.0}^{\Delta }$ exo, respectively.¹ The accomodative convergence/accomodation (AC/A) ratio, measured with the gradient method, was $2^{\Delta }∕1$ . The blur, break, and recovery of convergence at far distance were ${11}^{\Delta }$ , ${29}^{\Delta }$ , and $9^{\Delta }$ , respectively, and the break and the recovery of divergence were ${10}^{\Delta }$ and $6^{\Delta }$ . At the near distance, the values of the break and the recovery of convergence were ${29}^{\Delta }$ and ${19}^{\Delta }$ , and those of divergence were ${22}^{\Delta }$ and ${17}^{\Delta }$ . All measurements of fusional ranges were obtained using a rotary prism.¹ F.Z. was negative on the Worth four-dot test (both far and near distance) and had normal stereo-acuity (40″ of arc) measured with Wirt rings at $40\mathrm{cm}$ . On the Borish vecto-graphic near point card, he showed no fixation disparity.

3.3.

Stimuli

3.4.

Apparatus

Horizontal LE and RE movements were simultaneously recorded by an ET4 Infrared Eye Tracking System (AMTech, Weinheim, Germany). The sampling rate was $500\mathrm{Hz}$ . At best, the system has a horizontal resolution of $5\mathrm{arcmin}$ , and, as reported by the manufacturer, it is linear over a $20\text{-}\mathrm{deg}$ range. The output of the recording system is represented by the projections of the LE and RE lines of gaze on the PC screen (points ${\mathrm{S}}_{\mathrm{L}}$ and ${\mathrm{S}}_{\mathrm{R}}$ in Fig. 1).

3.5.

Procedure

The participant sat on a chair in front of the PC screen. A forehead rest, strap-band head rest, chin rest, and bite bar were used. He tried to avoid blinking during recording. The calibration procedure was carried out immediately before each experimental trial. The participant fixated the upper-left calibration point out of the $2\times 2$ calibration matrix while trying to align the dichoptic nonius lines. When alignment was perceived and stabilized, he pressed a key to signal the recording system to acquire that position as a reliable binocular calibration datum. Then he had to saccade to the next calibration point (upper right) and repeat the alignment procedure. The calibration sequence (upper left, upper right, lower left, lower right) was repeated twice. Immediately after calibration, the task was performed. The participant had to saccade to the far target. His task was to maintain fixation on the far target and press a key (connected to the recording system) to track from the moment when he was able to keep the nonius lines aligned. Then he had to switch his gaze to the near target. Again, he had to press the key to signal alignment on the near fixation target, then return to fixate the far target. Each sequence of shifts lasted about $30\sec$ (calibration excluded). No time constraint was applied, so the number of shifts from the far to the near target varied in each sequence (on average, approximately 16 shifts per sequence). Each sequence started with a new calibration. Overall, 3 sequences were performed.

3.6.

Analysis

The projections on the screen of the LE and RE lines of gaze (points ${\mathrm{S}}_{\mathrm{L}}$ and ${\mathrm{S}}_{\mathrm{R}}$ , respectively, in Fig. 1) were the output of the recording system. The ${\mathrm{S}}_{\mathrm{L}}$ and ${\mathrm{S}}_{\mathrm{R}}$ positions (sampled every $2\mathrm{ms}$ ) were averaged across a $200\text{-}\mathrm{ms}$ interval starting from $200\mathrm{ms}$ before the key press (an interval took into account the response time for key pressing after alignment). The horizontal FD was calculated by applying Eq. 14. Known terms of the equation were $PD$ $\left(6.0\mathrm{cm}\right)$ , $d$ $\left(60.0\mathrm{cm}\right)$ , and ${\mathrm{S}}_{\mathrm{L}}$ and ${\mathrm{S}}_{\mathrm{R}}$ , averaged separately (across time). Twelve fixations on the far target and (separately) 12 fixations on the near target were included in the analyses. Positive values of FD corresponded to eso-disparity (i.e., crossed lines of gaze); negative values corresponded to exo-disparity (i.e., uncrossed lines of gaze).