Many of the everyday tasks that we perform require accurate representations of spatial relations in our immediate environment: grasping a cup, threading a needle, or hammering a nail. Such geometric representations are based in part on binocular visual input, yet a variety of studies have suggested that binocular visual perception of object size, distance, and layout is typically distorted. Early experiments on size constancy (Blumenfeld, 1913; Hillebrand, 1902) led to the conclusion that binocular visual space is non-Euclidean (for reviews on empirical work, see Blank, 1978, and Indow, 1991 ). Based on the results of Hillebrand’s (1902) and Blumenfeld’s (1913) experiments, Luneburg (1947) proposed a model in which visual space is a Riemannian space of constant negative curvature (for reviews, see Roberts & Suppes, 1967, and Suppes, Krantz, Luce, & Tversky, 1989).

Recent direct tests of this model indicate that the assumption of constant curvature does not hold. Estimates of curvature varied when the task was performed at different distances from the observer (Koenderink, van Doorn, & Lappin, 2000) or when context objects were added to the scene (Cuijpers, Kappers, & Koenderink, 2001; Schoumans, Kappers, & Koenderink, 2002; Schoumans, Koenderink, & Kappers, 2000). These results lead to the conclusion that visual space may not have a single, global Riemannian geometry of any simple description. It also hints that the intrinsic structure of visual space varies with the tasks that are used to determine it.

If the local geometric structure of visual space differs from region to region, as Koenderink and colleagues conclude, then there is no basis for generalizing from one region of binocular visual space to another. A region of particular interest is the range of near space accessible to arm movements, where eye and hand can coordinate in carrying out a task (see, e.g., Hayhoe, Shrivastava, Mruczek, & Pelz, 2003). In this region, which we refer to as visual grasp space, errors in visual or motor estimates can be most punishing and, at the same time, the visual and motor systems can each be used to calibrate the other (Berkeley, 1709).

There has been little previous research on the geometric structure of visual grasp space. The performance of human observers during line bisection tasks suggests that visual grasp space is distorted. In these studies, a line is presented in the frontoparallel plane at viewing distances of 50 to 100 cm. Observers are instructed to perform a (one-dimensional) judgment and indicate the location (the bisection point) that splits a line into two equal parts. The majority of studies report leftward bisection errors (i.e., the perceived bisection point occurs to the left of the line’s center point), but a large number also report either nonsignificant leftward errors or a relatively high incidence of normal observers who demonstrate rightward bisection errors. In general, studies with normal observers demonstrate strong individual differences in bisection biases (Post, Caufield, & Welch, 2001; for a review, see Jewell & McCourt, 2000). At present it is not clear what the explanation is for these performance differences. The pattern of bisection biases varies with the details of the experimental methodology. Biases occur in studies using forced choice (McCourt & Olafson, 1997) or method-of-adjustment (MOA) (Jewell & McCourt, 2000), independent of whether or not the interval being divided contains a line (Bradshaw, Bradshaw, Nathan, Nettleton, & Wilson, 1986; Post, Caufield, & Welch, 2001).

In these studies, and in many earlier studies of the geometry of binocular space, eye fixation was not controlled or monitored. If distortions in binocular space vary with eye fixation, then the reported discrepancies between observers may be due to differences in eye-movement strategies. Haubensak (1970) had earlier offered such an explanation for the task-dependence observed in earlier studies of binocular visual space. He argued that the discrepancies between different geometric judgments observed in Hillebrand’s (1902) and Blumenfeld’s (1913) experiments might have resulted from a method-dependent artifact. He argued that fixation strategies differ between the setting tasks (distance vs. parallel judgments). If the geometric structure of binocular space changed with changes in fixation, then differences in fixation strategy could lead to apparently inconsistent settings. Along the same line of thought, Ehrenstein (1977) demonstrated that absolute versus simultaneous versus successive judgments of size constancy would lead to different curvature estimates of visual space. He suggested that the variety and inconsistency of curvature estimates obtained were caused by differences in the eye fixation strategies observers used for the different comparisons. Empirical studies along this line of reasoning have not been pursued further.

Independent of Haubensak’s ideas (1970), some bisection studies have asked whether observers perform bisection using particular scanning strategies and how the observed biases correlate with observed patterns of eye movements. In a forced-choice tachistoscopic line bisection task, mean gaze deviation from the line center was positively correlated with bisection error (McCourt, 2001). However, in another study, neither the median of fixation position nor the point of longest fixation was found to correlate with biases in bisection for normal observers (Barton, Behrmann, & Black, 1998). Studies in which observers’ scanning strategies were manipulated have found significant biases in line bisection. While biases in bisection vary under different viewing instructions within individual observers, these effects have failed to be explained by a single factor, but rather seem a combination of several factors, such as viewing distance, line length, starting position, and scanning direction (Bradshaw et al., 1986; Brodie & Pettigrew, 1996; Chokron, Bartolomeo, Perenin, Helft, & Imbert, 1998; Jewell & McCourt, 2000, Varnava, McCarthy, & Beaumont, 2002). It has also been argued that changes in line bisection under different viewing conditions may be due to a shift of attention along multiple spatial dimensions (Varnava, McCarthy, & Beaumont, 2002), or due to perceptual (“illusionary”) effects induced by visual context (Post et al., 2001).

If geometric judgments are affected by differences in fixation strategy, this effect should be largest for visual grasp space, where the range of vergence angles is large, and hence the vergence cue is most reliable and has a substantial role in the perception of absolute distance (Berkeley, 1709; Collett, Schwarz, & Sobel, 1991; Cumming, Johnston, & Parker, 1991). For example, Tresilian and coworkers found that within near binocular space (<1 m) the weight given to vergence is increased (Tresilian & Mon-Williams, 2000; Tresilian, Mon-Williams, & Kelly, 1999). Similarly, Viguier, Clément, and Trotter (2001) observed a linear relationship between (distorted) perceived distance and the actual distance of a target when the perceived distances were expressed as vergence angles, and concluded that vergence is used to estimate target distance in near binocular space (for conflicting views, see Logvinenko, Epelboim, & Steinman, 2001).

In our study, we address three issues. We first seek to characterize the patterns of distortion not simply in the fronto-parallel plane but in three dimensions (Experiment 1). Second, we compare the pattern of distortions found for two different bisection tasks to directly assess the task dependence of bisection in visual grasp space (Experiment 2). Third, we test whether and to what extent distortions in bisection judgments depend on the observer’s eye fixation point (Experiment 3).

In Experiment 1, observers were instructed to bisect the imaginary line between two points in space using a three-dimensional method of adjustment. The chosen stimulus configurations followed a paradigm introduced by Blank (1958, 1961), and allowed for an explicit test of the congruence between visual grasp space (of possible constant curvature) and Euclidean physical space. In Experiment 2, we study whether observers choose the same point (and prefer it over the Euclidean bisection point) in a two-alternative forced-choice (2AFC) paradigm. In both of these experiments, observers performed the bisection judgment under free viewing conditions. In Experiment 3, we determine whether bisection is affected by eye fixation location. In this experiment, observers were instructed to bisect the imaginary line using 2AFC while maintaining fixation at specified points in the stimulus configuration, and we compare the judgments for different fixation points.

Throughout this work and the literature preceding it, experimenters find themselves testing null hypotheses, such as “settings are independent of fixation” or “binocular space is Euclidean.” It is important to note that it is implausible a priori for such hypotheses to be true if measurements are taken to arbitrary precision. A failure to reject is likely due to a lack of statistical power. With enough data, it is virtually certain that such hypotheses can be rejected. No coin is ever exactly fair, but a large number of coin tosses may be needed to determine whether it is biased ever so slightly toward heads or toward tails. What is of interest here is the relative magnitudes of the failures of the various models under consideration and their consequences for biological vision. We return to this point in the discussion, after we have estimated these magnitudes.

In Experiment 1, we examine bisection in three-dimensional space under natural viewing conditions (and without a response time limit). Observers made three-dimensional adjustments and did not receive instructions where to fixate while performing the task.

Two Sony Trinitron Multiscan G500 monitors, positioned on either side of the observer, were viewed using two half-silvered mirrors, one for each eye, forming a Wheatstone stereoscope (Figure 1). The partial transparency of the mirrors facilitated spatial calibration of the monitors (described below) but played no other role in the experiment. These monitors are close to physically flat (less than 1 mm of variation across the extent of the screen). The optical distance from each eye to the center of the corresponding monitor was 70 cm. From this distance, the central region of each screen, used to display our stimuli, subtended 20 × 20 deg.

Observers were positioned in a chin rest and were asked to keep their heads still. No head restraint was imposed. The apparatus was contained in a large box whose interior was covered in black flocked paper (Edmund Scientific), an efficient light-absorbent surface. The observer could see only the points defining the stimulus, apparently floating in front of him or her against a black background. The task of the observer was to move a point in space until it appeared to bisect the line segment defined by two other points. Each point was a trivariate Gaussian “blob” of light that could be positioned in space with high resolution. Directly in front of the viewer, at a distance of 70 cm, this resolution was 0.07 mm in the horizontal and vertical directions and 0.14 mm in depth, corresponding to 21-s visual angle in the vertical and horizontal directions and 42 s of disparity resolution. This resolution is small compared to observers’ setting variability in these tasks. The anti-aliasing methods used to present these points in stereo are described by Warren, Maloney, and Landy (2002) and are based on the work of Georgeson, Freeman, and Scott-Samuel (1996).

The observer calibrated the apparatus spatially before each experimental session. Using only the left eye, the observer first viewed a 4 × 5 array of points on the left monitor superimposed on a physical reference target by means of one of the half-silvered mirrors. The calibration reference target was a 4 × 5 array of points on a rigid planar surface placed 70 cm in front of the observer. The observer moved each point separately until it appeared to lie on top of the corresponding physical reference dot. This process was then repeated for the right eye. These data were used to calibrate the placement of dots from each eye’s view as described by Warren et al. (2002).

On each trial, the observer saw two fixed points and an adjustable point. The fixed points were two of the three vertices of an equilateral triangle with 14 cm sides. The triangle lay in the horizontal plane through the two eyes, centered on the midline (Figure 2). There were three bisection conditions defined by the three possible vertex pairs: left, right and back. This allowed us to examine the frontoparallel bisection judgment that most other studies have used as well as bisections of points varying in both azimuth and depth. The adjustable point was initially positioned at a random location within a sphere of radius 4 cm centered on the Euclidean bisection point.

In each trial, the observer moved the adjustable point in three dimensions until it appeared to bisect the invisible line segment joining the two visible fixed points. Observers used eight buttons to control the experiment. Six of these, organized in pairs, were used to move the adjustable point parallel to each of the three Cartesian axes, x (left-right), y (front-back), and z (up-down). Pressing one key of the pair moved the point one way along the axis, pressing the other moved it in the opposite direction. A seventh button controlled the speed at which the point moved. At the start of a trial, the control program permitted “quick” movement of the point – each key press displaced the point by approximately 0.5 mm. When the observer judged that the adjustable point was near the bisection point, they pressed a seventh button, the speed toggle, which allowed them to move the point with greater precision (at the limit of resolution of the apparatus) until they were satisfied with their setting. A final press of an eighth button recorded the observer’s setting and triggered the next trial. Observers were encouraged to move their eyes across the vertex pair before completing their judgment.

The three bisection tasks were interleaved, and the observer carried them out in the order left, right, back, repeatedly, until completing 40 settings for each of the three tasks split over two sessions of 40-min duration each.

Four observers completed the two sessions. Three of the four were experienced psychophysical observers who were unaware of the purpose of the experiment. The remaining one, JT, was the first author.

The results are summarized in Figure 3 and Table 1. Figure 3 shows the mean settings and an ellipse indicating the setting variability projected onto the xy plane (i.e., seen from above). The variance ellipses indicate ±2 SD of the setting. Table 1 shows the mean deviations (±1 SEM) of the settings from the Euclidean bisection point in the x, y, and z directions, as well as the Weber fraction (as a percentage) for the three-dimensional adjustment. Negative mean deviations from the Euclidean bisection point indicate biases leftward (x), away from the observer (y), and downward (z).

We computed the deviation of observers’ bisection settings from the Euclidean bisection point (Table 1). For some of the observers, these deviations were large compared to setting variability. All observers’ had bisection settings that deviated significantly from the Euclidean bisection point (t test, p < .05). The patterns of deviations differ from observer to observer and are reproducible from observer to observer (see also Experiment 2, which was performed on the same observers two weeks later, and Experiment 3, which JT performed 11 months after Experiment 1). Deviations were symmetric with respect to the stimulus vertex pair in the xy plane, except for observer JT whose deviations exhibited an overall rightward bias. In the vertical direction, the bisection settings were biased upward by approximately 0.1–0.6 mm (Table 1).

In summary, bisection settings were stable and consistent throughout the experiment, were often significantly different from the Euclidean bisection point, and differed significantly in direction across the four observers.

In this experiment, observers bisected an invisible line using a three-dimensional method of adjustment. Although no specific instructions regarding fixation were given, observers were encouraged to move their eyes across the vertex pair and to select the bisection point that appeared most satisfactory across these varying fixations. Under these conditions, the recorded bisection settings were stable between the two sessions for all four observers.

The back vertex pair consisted of two points displayed 14 cm apart in the frontal plane (symmetric with respect to the line of sight). In this vertex pair, two observers, JT and HB, exhibited rightward bisection biases, while the other two observers, IO and KB, did not show a bisection bias in the x direction. A leftward bisection bias, often observed in line bisection experiments in the frontal plane, did not occur for any of the observers with this vertex pair. However, in contrast to standard line bisection experiments in which observers are instructed to bisect a line using a one-dimensional judgment (along the line), in our experiment observers bisected the frontal vertex pair using a full three-dimensional adjustment. Being able to place the bisecting point anywhere in space, and not restricted to a position along the line, three observers chose settings significantly deviating from the bisecting line, either in depth (observers IO and JT), or in the vertical direction (observers JT and KB).

As mentioned in the “Introduction,” biases in perceived geometry that differ from Euclidean predictions have been attributed to an underlying Riemannian geometry of perceptual space with nonzero curvature (Indow, 1991). Empirical studies in this vein, but typically carried out at greater distances from the observer than the present study, have found different values of the curvature parameter k, both positive (indicating a spherical geometry) and negative (indicating a hyperbolic geometry). Curvature estimates have even been found to vary within individual observers if measured at different locations in space (Koenderink et al., 2000) or when context objects are added to the scene (Cuijpers et al., 2001; Schoumans et al., 2000, 2002). In agreement with these previous results (which involved distances beyond grasp space), our results contradict a geometric interpretation of visual space, both qualitatively and quantitatively. Were the biases found in our study a result of a non-Euclidean geometry of constant curvature, we would expect all the bisection settings to be outside the triangle (for a positive curvature) or all inside (for a negative curvature). This is not true of most of our subjects, even in the small region of visual grasp space immediately in front of the observer. Second, given an empirically estimated value of curvature from a previous study, one can calculate the bias expected in the present experiment, which uses an inter-point distance that is far smaller. The predicted biases are much smaller than those we found. We conclude that our results cannot be attributed to an underlying Riemannian geometry of constant curvature (compare Cuijpers, Kappers, & Koenderink, 2001, 2002; Koenderink, van Doorn, & Lappin, 2000, 2003; Todd, Oomes, Koenderink, & Kappers, 2001).

In the second experiment, we test whether the bisection biases found in Experiment 1 are independent of the method of judgment employed. We tested whether observers chose the same bisection point in a 2AFC task as they did in the MOA task in Experiment 1. In this experiment, the observer had to decide which of two points, presented sequentially, appeared closer to the perceived bisection point. In particular, we tested whether the mean bisection setting point of Experiment 1 (the MOA bisection point) is preferred over the Euclidean bisection point when presented as a forced choice.

The apparatus was the same as in Experiment 1.

Each observer was presented with one of the three vertex pairs that they viewed in Experiment 1 (the left vertex pair for observers HB, JT, KB, and the right for IO). On each trial, two candidate bisection points were displayed briefly, one after the other. The two points were drawn from a set of points that fell on an invisible line segment connecting the Euclidean bisection point and the MOA bisection point (Figure 4). A sequence of 7–11 equally spaced points along this line was chosen as follows. The SD of the MOA settings along that line was computed. Points were spaced using the largest possible increment, smaller than 1 SD, such that the evenly spaced sequence included the Euclidean and the MOA bisection points. Figure 4 shows the projection of the sequence of points onto the xy plane. Because the MOA bisection point could have differed from the Euclidean bisection point in the z coordinate, the sequence of points may also have included increments in the z direction. See Table 2 for details.

In each trial, the observer was presented with two points, flashed for 150 ms each, and separated by an interval of 1 s. Subjects indicated by mouse buttons which of the two appeared to be closer to the bisection point. Observers were not specifically instructed where to fixate in the stimulus display.

All pairwise comparisons of the candidate bisection points were performed in randomized order. Observers HB and JT performed 10 repetitions per pairwise comparison, for a total of 450 (HB) and 550 (JT) trials; observers IO and KB performed 20 repetitions per comparison for a total of 420 trials. Pairwise comparisons were balanced with respect to the order of presentation of the two points. The experiment was completed in two sessions of approximately 35 min each.

The same four observers completed the two sessions as in Experiment 1, which were run two weeks after Experiment 1.

The results are summarized in Figure 5. All four observers preferred the MOA bisection point over the Euclidean bisection point.

We estimated the preferred bisection point as follows. For each observer, the raw data constituted a “probability preference matrix” [p_ij] based on the actual observer’s data set (i.e., the 10 or 20 repetitions per point pair ij). The preference probability p_ij indicates the proportion of occasions in which this observer chose point i over point j in a pairwise comparison.

We used Efron’s Bootstrap to provide an error estimate (Efron & Tibshirani, 1993). The experiment was simulated using the same number of repetitions of each pairwise comparison, and with random choices based on the p_ij values. In each simulation, the modal value was chosen to represent the most preferred point. A distribution of preferred points was generated by performing 5,000 simulation runs. Figure 5 displays the bootstrap estimates of the mean and SD of the simulated 5,000 modal values, as well as the mean and SE of the MOA judgments from Experiment 1 (see also Table 3).

For all four observers, the preferred bisection point in the 2AFC task was several SDs away from the Euclidean bisection point. We tested whether the MOA and 2AFC tasks yielded the same bisection point. For the statistical analysis, a z value was computed by dividing the difference of mean MOA setting and mean preferred bisection point in the 2AFC task by the combined variance estimate of the MOA setting and the bootstrap estimate of the variance in the 2AFC task. (To reduce the probability of a type-II error, the statistical test was performed based on a z value instead of testing a t value corresponding to the finite number of degrees of freedom in the MOA judgment.) Table 3 contains the resulting p values and indicates that preferred bisection points in the 2AFC task did not differ significantly from the MOA settings. We therefore conclude that observers select a bisection point independent of the method of judgment.

The results of Experiments 1 and 2 show that bisection is performed independent of the methods used (MOA or 2AFC). However, observers’ settings deviate significantly from the Euclidean bisection point. Also note that Experiment 2 was performed two weeks after Experiment 1 on the same observers. This suggests that the observed biases are not caused by the specific demands of a particular type of judgment and remain stable across time.

Note that observers did not receive instructions as to where in the stimulus configuration to fixate. However, all four observers reported afterward that during the brief stimulus display they fixated in the area of the bisection point. This is in agreement with a study on the patterns of eye fixation during line bisection (Barton et al., 1998) that found that subjects mainly scanned the center of the line symmetrically, and seldom fixated near the ends of the lines.

Experiment 3 was performed to test whether the perceived bisection point varies with changes in fixation. Observers were instructed to bisect the imaginary line defined by two visible points using 2AFC. They did this while maintaining fixation at specified locations in the stimulus. On each trial, they were instructed either to fixate one of the two visible points of the vertex pair, or the invisible bisection point. While they fixated, a point was briefly flashed near their MOA bisection point. There were three displacement conditions. In the left-right condition (x direction), the point was either to the left or right of the MOA bisection point. Observers indicated whether the point was left or right of the bisection point. Back-front (y direction) and up-down (z direction) displacement conditions were run as well.

The apparatus was the same as in Experiments 1 and 2.

The stimulus vertex pairs (the two visible points) were the same as in Experiment 1. Observer JT performed the left and back conditions, and observer KL performed only the left condition.

Observers performed a 2AFC judgment of the direction in which a briefly flashed point deviated from the bisection point. Judgments were performed along the three Cartesian axes (x: left vs. right, y: behind vs. in front of, and z: above vs. below) in separate blocks of trials.

For both observers, we first measured the MOA bisection point. Based on 20 settings for each vertex pair, the mean and standard deviation of the MOA settings were estimated along the x, y, and z directions, separately for each observer and vertex pair. Then, a series of 9 points was chosen in each Cartesian direction containing the MOA subjective bisection point plus 8 points whose distances from the MOA bisection point were chosen based on the MOA SD in steps of ±0.5 SD, ±1 SD, ±2 SD, and ±4 SD. As the SD in the z direction was very small (< 0.3 mm), grid points were placed with a spacing of ±1 SD, ±2 SD, ±3 SD, and ±4 SD in the z direction.

Each trial started with the presentation of the vertex pair. A color signal indicated the point of fixation. If one of the two corner points turned red (for 500 ms), the observer was instructed to fixate the indicated point and maintain fixation there until after display of the candidate bisection point to be judged. If both corner points turned blue (also for 500 ms), this indicated that the observer should fixate near the bisection point (although there was no fixation point displayed in that region). At 750 ms after the fixation signal, the candidate bisection point was flashed for 150 ms. Observers indicated by pressing a mouse button whether the presented stimulus was to the left or right of (judgment in the x direction), behind or in front of (judgment in the y direction), or above or below (judgment in the z direction) the bisection point.

Trials were blocked by the direction to be judged (x, y or z) and by vertex pair (for observer JT) in blocks of 135 trials. Fixation conditions were randomized within each block. Observers completed 50 judgments per point and fixation, amounting to a total of 4,050 trials per vertex pair. The experiment was completed in 13 (KL) and 25 (JT) sessions of approximately 30 min each.

Two observers completed the experiment. One was an experienced psychophysical observer who was unaware of the purpose of the experiment and had not participated in the previous two experiments. The other, JT, was the first author.

Data were analyzed by fitting a cumulative Gaussian distribution function to the frequency distribution of left (x), behind (y), or below (z) judgments using the maximum-likelihood method as implemented by Wichmann and Hill (2001a, 2001b). The level corresponding to 50% (the point of indifference) was taken to indicate the bisection point.

Figure 6 shows the raw data and the fitted psychometric functions. In some cases, the curves appear to be displaced relative to one another, indicating different biases for the various fixation conditions (see Table 4 for deviations of the point of indifference from the MOA setting under different fixations). In addition, the curves for the condition in which the bisection point is fixated are steeper, indicating that observers’ bisection judgments were more accurate when fixating near the point of bisection.

A bootstrap analysis was performed to test whether bisection points varied significantly with change in fixation. For a given observer, condition (left or back) and axis (x, y, or z), we calculated the maximum difference Δ between the measured bisection points based on the three fixation conditions. Next we assumed that the underlying bisection points were identical, differing only in their measurement variability (the variance of the estimated 50% point of the fit). 50,000 times, we drew three random values for each of the three bisections based on identical means and the measured variabilities, and calculated a new value of Δ. The p value reported in Table 5 is based on the percentile in the distribution of Δ values corresponding to the measured value of Δ. For each subject and condition, at least one of the three axes of bisection displays a significant difference across fixation conditions.

In this experiment, observers performed 2AFC bisection judgments along the three Cartesian axes while maintaining fixation at either of the two corner points of the vertex pair or while fixating the middle of the invisible line between the two points. Results indicated that bisection points differed significantly under the different fixation conditions.

Judgments were more accurate (i.e., yielded steeper psychophysical functions) for fixation in the middle of the vertex pair. This result is consistent with the findings in Experiment 2 where observers were free to move their eyes during the 2AFC judgment and had reported that they had fixated the middle of the vertex pair throughout the experiment (see also Barton, Behrmann, & Black, 1998).

We also note that our conclusions do not depend on whether observers fixated precisely where they were told to fixate. Suppose, for example, that the observers had simply ignored the different fixation instructions and had fixated at some preferred, default location. Then we could not expect to find the differences between fixation conditions that we did find. We are confident that observers did change fixation in response to instructions and that, due to the brevity of the stimuli, they had little opportunity to change fixation once the trial had begun.

The pattern of results we have found may be summarized as follows. Observers in Experiment 1 made bisection settings in three dimensions in visual grasp space. These MOA settings were significantly different from the Euclidean bisection points and differed from observer to observer. Their settings were not consistent with models of binocular space of constant curvature. In Experiment 2, we verified that observers select these same bisection points given a forced-choice between their own MOA settings from Experiment 1 and other nearby points, including the Euclidean bisection point. For each of four observers, we could not reject the hypothesis that the observer preferred his or her own MOA setting to any of the alternative points provided.

In both of these experiments, observers were free to adopt any fixation strategy they chose. In Experiment 3, observers were asked to judge whether points fell to the left or right (above or below, front or back) of the subjective bisection point while fixating a specified point in space. Observers were asked to fixate either the invisible bisection point or one of the end points of the vertex pair to be bisected. We found significant differences in bisection performance in different fixation conditions.

In Experiment 3, bisection points under different fixations were estimated using the results of 450 2AFC judgments per condition. With this amount of data, we can reliably detect very small changes in bisection judgments with changes of fixation and estimate their magnitudes. As noted in the “Introduction,” the key question is not whether there are such changes, but what their magnitudes are relative to other measures of visual performance.

We can, for example, ask whether the observer would likely be able to detect the inconsistency in his or her own bisection judgments with changes in fixation. To address this question, we compared bisection points in Experiment 3 under different fixations to a measure of just noticeable difference (JND) for a single trial derived from the slope of the psychometric function. For each condition, the JND was estimated as the interval between the 25% and 75% level of the fitted psychometric functions (Table 4). Figure 7 displays the bisection points under different fixations and the corresponding JNDs. It is clear from Figure 7 that the estimated bisection points under different fixations are within a JND or so of one another in all conditions, for all observers. Although bisection points may differ significantly from one fixation to another, these differences are likely to go undetected as the observer shifts fixation. That is, an observer will not often reject his or her own previous bisection setting or judgment as a consequence of change in fixation.

We can also compare the differences in bisection judgment with change in fixation (Experiment 3) to the observers’ uncertainty in MOA settings (Experiment 1). In most, but not all cases, bisection points measured under the different fixation conditions fell within the range of measurement uncertainty of the method of adjustment (Figure 7). In the MOA condition, observers had been instructed to select the setting that was most satisfactory under free viewing conditions. It is therefore possible that the setting uncertainty in this task comprises the small differences in bisection points associated with the various possible fixations observers may have adopted from trial to trial.

Finally, we can address whether the geometrical structure of visual grasp space varies with changes in fixation: Are the deviations from the Euclidean bisection point that we found in Experiment 1 large compared to the differences induced by changes in fixation (Experiment 3)? Comparing Tables 1 and 5 for the one subject who ran in both experiments, differences between perceived and Euclidean bisection points for observer JT are 2 to 8 times larger than the differences detected between bisection points under different fixations (except for the z direction in the back configuration). We therefore reject the conjectures of Haubensak (1970) and Ehrenstein (1977). Bisection performance does depend on fixation. However, the effect of fixation location on bisection is not large enough to be detected reliably by the observer moving his or her eyes during a judgment, and the effect is not large enough to explain the discrepancies observed between bisection judgments and a model based on Euclidean or, more generally, a Riemannian geometry.

In near space, in particular, it has been suggested that calibration processes assure the geometric consistency of judgments and support eye-hand coordination (Berkeley, 1709; von Helmholtz, 1867, 1878). Recent work emphasizes how discrepancies in geometric judgments can be used as the input to a calibration algorithm (Maloney, 1996; Maloney & Ahumada, 1989). In these algorithms, perceived changes in simple geometric judgments with changes in fixation and head position are the trigger driving visual calibration to reduce or eliminate such inconsistencies. Thus, rather than attributing bisection biases to a particular theory of the structure of visual space, we suggest that these biases reflect a tolerance for miscalibration of space within the individual. As such, these biases can be idiosyncratic, varying across individuals and regions of space. While only suggestive, it is of interest that the inconsistency of bisection judgments under changes of fixation is somewhat smaller than measures of visual positional accuracy on a trial-by-trial basis (MOA setting variability and JND), but only slightly so.

This research was supported by grant EY08266 from the National Institutes of Health and grant RG0109/1999-B from the Human Frontiers Science Program. J.T. was also funded by the Deutsche Forschungsgemeinschaft (Emmy-Noether Programm). Finally, we thank Katja Dörschner for creating the icon. Commercial relationships: none.