Spatial Biases and the Haptic Experience of Surface Orientation

TL;DR: The two main purposes of this chapter are to review past evidence for a systematic spatial bias in the perception of surface orientation (geographical slant), and to report two new experiments documenting this biases in the manual haptic system.

Abstract: The two main purposes of this chapter are to review past evidence for a systematic spatial bias in the perception of surface orientation (geographical slant), and to report two new experiments documenting this bias in the manual haptic system. Orientation is a fundamental perceptual property of surfaces that is relevant both for planning and implementing actions. Geographical slant refers to the orientation (inclination or pitch along its main axis) of a surface relative to the gravitationally-defined horizontal. It has long been known that hills appear visually steeper than they are (e.g., Ross, 1974). Only recently has it been documented that (1) there is also bias in the haptic perception of surface orientation (Hajnal et al., 2011), and that (2) similar visual and haptic biases even exist for small surfaces within reach (Durgin, Li & Hajnal, 2010).

Summary

1. Introduction

• The two main purposes of this chapter are to review past evidence for a systematic spatial bias in the perception of surface orientation (geographical slant), and to report two new experiments documenting this bias in the manual haptic system.
• Geographical slant refers to the orientation (inclination or pitch along its main axis) of a surface relative to the gravitationally-defined horizontal.
• Only recently has it been documented that (1) there is also bias in the haptic perception of surface orientation (Hajnal et al., 2011), and that (2) similar visual and haptic biases even exist for small surfaces within reach (Durgin, Li & Hajnal, 2010).
• Following the presentation of the experimental results the authors will discuss issues of measurement in perception – especially pertaining to the interpretation of verbal reports, and conclude with a discussion of functional theories of perceptual bias in the perception of surface orientation.

2. Spatial bias in the perception of orientation: Surfaces within manual reach

• Durgin, Li and Hajnal (2010) reported a series of studies of a bias they called the “vertical tendency” in slant www.intechopen.com Haptics Rendering and Applications 76 perception.
• When Durgin, Li and Hajnal asked participants to make estimates of the geographical slants of wooden surfaces within reach, they found that that they got approximately the same bias function whether the surfaces were at eye level (so that “frontal” and vertical coincided) or viewed with gaze declined by about 40°.
• The estimates shown in Figure 1 are based on verbal/numeric estimates of orientation relative to horizontal, but the bias observed cannot be due to verbal coding.
• Thus, it appears that several different perceptual representations of pitch contain a bias that expands the scale of differences near horizontal while compressing the scale near vertical.

3. Bias in the perceived orientation of locomotor surfaces: Hills and ramps

• A difficulty with this view is that perceived hill orientation decreases as one approaches a hill (Li & Durgin, 2010), and nearer portions of hills appear shallower than farther portions (Bridgeman & Hoover, 2008).
• In fact, as the authors will discuss below, there is continuity between orientation biases they have measured for small, near surfaces and those measured by Proffitt et al. for hills (Li & Durgin, 2010).
• Proffitt further proposed that physical actions, such as stepping, were controlled by an unbiased perceptual representation (vision for action) that was contrasted with the exaggerated representation available for long-range cognitive planning.
• This view has since been challenged by studies of the haptic perception of locomotor inclines.

3.1 Haptic bias in the perceived orientation of locomotor surfaces

• Hajnal et al. (2011) asked participants to step onto ramps that they could not see.
• (They were either wearing a blindfold or an occluding collar that blocked their view of the floor.).
• The data are reproduced in Figure 4, along with a photograph of the experimental situation.
• For somewhat steep ramps, the haptic exaggeration of perceived slant was even greater than the visual exaggeration observed when the same participants judged the orientations of the ramps when looking at them afterward.
• To rule out the possibility that the haptic exaggerations were learned from calibrating haptic experience to visual experience, Hajnal et al. (2011) also tested a population of four congenitally blind individuals using verbal report.

3.2 Proprioceptive bias in the perceived orientation of locomotor surfaces

• Proprioceptive error in the perceived declination of gaze was first reported in a study of downhill slant perception: Li and Durgin (2009) observed that standing back from the edge of an outdoor downhill surface made it appear steeper than when standing closer to the www.intechopen.com Haptics Rendering and Applications 80 edge.
• Indeed, for a steep hill, the maximum perceived orientation seemed to occur when standing far enough back from the edge of the hill that one’s line of gaze was nearly coincident with the surface of the hill.
• Fig. 4. Verbal and proprioceptive (hand gesture) estimates of the haptically-perceived orientation of a ramp while standing on it, blindfolded (from Hajnal et al., 2011).
• Proprioceptive points are displaced to show the SEMs.
• Deducing from these observations that the perceived direction of gaze might itself be distorted Li and Durgin (2009) tested this directly by asking people to look at targets at various declinations out of upper-floor windows and estimate the downward pitch of their gaze.

3.3 Continuity between visual biases for near and far surfaces

• Across a number of studies, Li and Durgin (2009, 2010; Durgin & Li, 2011a) have found systematic evidence suggesting that the visual perception of slant also has a gain of 1.5 in the low end of the geographical slant range.
• Predictions of outdoor visual data from Proffitt et al. (1995) based on a model with only one free parameter.
• By setting the slant gain to 1.5 and assuming an overall intercept of 0°, the authors can reduce the model to a single free parameter, based on the multiplier of log viewing distance.
• This analysis provided by Li and Durgin (2010) shows how the apparent discrepancy between the perceived slants of hills and of near surfaces may be due to differences in viewing distance.
• The most intriguing observation the authors can make about this concerns the discrepancy between the haptically perceived slant of the 16° ramp (~35°) and the visually perceived slant of that same ramp (~23°).

4. Problems with measuring perceived slant with haptic matching tasks

• One current controversy in the study of slant perception concerns a popular method of assessing perceived slant.
• Durgin, Hajnal, Li, Tonge and Stigliani (2010) reasoned that if palm boards were assessing accurate motor representations of space, then they ought to be particularly accurate for www.intechopen.com Spatial Biases and the Haptic Experience of Surface Orientation 83 matching near surfaces with which the hand could actually interact.
• The hand was occluded from vision in all cases.
• Because the axis of the palm board was near the wrist, the wrist had to be the principal joint for adjusting the palm board.
• First, evidence of surprising accuracy between palm board matches and hills has turned out to be spurious.

4.1 Calibration between proprioception and the visual experience of slant

• The function shown in Figure 6 for free-hand manual gestures was obtained with the same set of surfaces used to obtain the function in Figure 1 for verbal estimates of slant.
• Nonetheless the verbal estimates show a great deal of bias (exaggeration of the deviation from horizontal), whereas the manual gestures appear to be well-calibrated.
• That is, the perceived orientation of the hand ought to match the perceived orientation of the surface.
• In support of this view, Li and Durgin have found that perceived hand orientation during free-hand gestures follows the same function as the verbal pattern shown in Figure 1.

4.2 An apparent discrepancy in the calibration account

• So far the authors have suggested that hand gestures are calibrated to near surfaces, but are not calibrated for far surfaces (which seem steeper).
• There is one apparent exception to this proposed guiding rule.
• They collected verbal estimates of perceived slant.
• Given the calibration account (based on the potential for shared visual and haptic experience in normal life) the authors should expect that haptic exploration of a surface by dynamic touch would reveal the same kind of spatial bias that is evident in vision and in static haptic contact .
• In the left panel, the linear fit line originally plotted by Hajnal et al. is shown.

5. Experiment 1: Numeric estimation of real surface orientation in depth assessed by dynamic touch with the tip of the index finger

• The main question of the present experiment is whether the haptic perception of surface orientation (geographical slant in the pitch axis relative to the observer) by dynamic touch will show the same kinds of spatial bias documented in vision by Durgin, Li and Hajnal (2010).
• As noted above, it is not clear that the linear fit they plotted is better justified by their data than a cubic fit, like that shown in Figure 7.
• Moreover, examination of the raw data of Hajnal et al. suggested that participants relied nearly exclusively on angular estimates that were multiples of 5.
• This may have contributed to distorting the lower end of the range.
• Finally, because Hajnal et al. did not constrain their participants’ exploratory strategies, it is possible that the observed function was less exaggerated in some places because of a tendency for oblique paths of travel along the slanted surface.

5.1 Method

• All experimental procedures were conducted in accord with the ethical standards of the American Psychological Association and approved by a local institutional review board.
• Participants made numeric estimates of the slant of surfaces explored haptically.

5.1.1 Participants

• The participants were 20 Swarthmore College undergraduate students (13 female) who participated in partial fulfilment of a course requirement.
• Half were assigned to the horizontal coding condition and half to the vertical coding condition.

5.1.2 Apparatus

• The haptic surface was a varnished wooden board mounted on a mechanical adjustable slant device (see Li & Durgin, 2009).
• The center of the surface was about 113 cm from the floor.
• Participants stood in front of the apparatus wearing a blindfold (a plush sleep mask) throughout the experiment.

5.1.3 Design

• Participants were assigned in alternation to the vertical or horizontal coding condition.
• Following the practice of Durgin, Li and Hajnal (2010), half the participants gave verbal estimates relative to horizontal and half gave estimates relative to vertical so that spatial biases could be distinguished from verbal biases.
• Random orders were generated in advance for each participant.
• Spatial Biases and the Haptic Experience of Surface Orientation 87.

5.1.4 Procedure

• Participants were shown the apparatus with the surface in the horizontal position and the procedure was explained to them prior to signing an informed consent form.
• Before each trial, the surfaces were set to the intended orientation manually using pre-set positions by the experimenter who then told the participant to explore the surface.
• No time limit was specified for exploration.
• Half were instructed that vertical was 0° and horizontal was 90°.
• Trend lines are best fitting cubic polynomials.

5.2 Results

• Mean estimates were computed for each presented orientation by condition.
• Figure 9 shows the estimates for each condition.
• It can be seen that the spatial bias was in the same direction in each condition inasmuch as participants overestimated deviations from horizontal and underestimated deviations from vertical.
• The functions are strikingly similar, as predicted www.intechopen.com Haptics Rendering and Applications 88 by the calibration hypothesis.
• Both functions seem to reflect a common underlying spatial coding bias.

5.3 Discussion

• Using real surfaces with a demarcated axis of haptic exploration, the authors sought to extend the methods used by Durgin, Li and Hajnal (2010) to the haptic domain.
• The authors results indicate a close correspondence between visual and haptic spatial biases in the peception of orientation.
• The authors results are somewhat at variance with those of Hajnal et al. (2011).
• Because Hajnal et al. did not constrain the path of digital exploration, it is possible that participants tended to explore their surfaces along a somewhat oblique (and therefore less steep) axis.
• The authors data are consistent with the proposal that there is a trend for there to be calibration between visual and haptic representations of 3D surface orientation.

6. Experiment 2: Horizontal/vertical bisection point for surface orientation in depth assessed by dynamic touch with the tip of the index finger

• To avoid verbal biases, Durgin, Li and Hajnal (2010) used a bisection task in which they presented surface visually and asked participants to indicate whether the surface was closer to vertical or to horizontal.
• They reported a mean visual bisection of point of 34° from horizontal.
• In fact the cubic fit to their verbal data predicted that the 45° point would have been at 36.3° in the visual case, and it seems likely that verbal reports tend to slightly underestimate the magnitude of the actual spatial bias (see also Durgin & Li, 2011a, 2011b).
• The present experiment simply replicated the bisection procedure of Durgin, Li and Hajnal for the haptic case.
• Spatial Biases and the Haptic Experience of Surface Orientation 89.

6.1.3 Design

• Each participant gave responses to individual stimuli selected from an up-down staircase procedure.
• There were 10 blocks of 6 trials each in which two trials from each of three staircases were randomly interleaved.
• That is, if the presented orientation was deemed closer to vertical, the next orientation presented by that staircase was 18° lower, and if the presented orientation was judged closer to horizontal, the next presented orientation was 18° higher.
• The three staircases together sampled orientation space with a resolution of 6° and approximated a method of constant stimuli that was centered on the apparent bisection point.

6.1.4 Procedure

• Participants were shown the apparatus with the surface in the horizontal position and the procedure was explained to them prior to signing an informed consent form.
• Participants were shown where to stand (directly in front of the apparatus) and then asked put on the blindfold.
• Before each trial, the surfaces were set to the required orientation manually by the experimenter according to a computer instruction.
• When the participant gave the forced choice response (“closer to vertical” or “closer to horizontal”), the experimenter pressed either the up-arrow key or the down arrow key on a keyboard, causing the computer to record the trial and update the staircase.
• The computer then gave instruction to the experimenter concerning the orientation of the next stimulus.

6.2 Results

• The responses for each participant were fitted with a logistic function and the subjective bisection point was calculated for each psychometric function as the point at which participants were equally likely to respond that the surface was closer to vertical and that it was closer to horizontal.
• The average subjective horizontal/vertical bisection point was 31.2° (SEM = 2.0°) from horizontal.
• Li and Hajnal (2010), this difference was not statistically reliable.

6.3 Discussion

• Using real surfaces with a clearly-demarcated axis of haptic exploration, the authors sought to extend the bisection method used by Durgin, Li and Hajnal (2010) to the haptic domain.
• The authors results are consistent with a close correspondence between visual and haptic spatial biases in the perception of orientation.
• The haptic results are also quite similar to the average perceived horizontal/vertical bisection point (31°) measured by Durgin and Li (2011a) for perceived gaze declination.
• In other words, across a variety of modalities (proprioception of gaze declination, visual perception of 3D surface orientation in depth and haptic perception of 3D surface orientation in depth) the perceived bisection point between horizontal and vertical is very close to 30° from horizontal.

7. A descriptive model of the slant bias function for manual reaching space

• The main purpose of their present study has been to clear up an apparent discrepancy in the coding of near-space orientation.
• That is, for manual reaching space, there had seemed to be a discrepancy between the dynamic touch results of Hajnal et al. (2011) and the static haptic results of Durgin, Li and Hajnal (2010).
• The present results support the idea that the same bias exists for dynamic touch as has been found for visual slant perception and static haptic slant perception.
• The bias function found previously for the evaluation of visual slants and for haptic slants experienced by static contact has now been replicated for dynamic touch by fingertip.
• In combination with recent evidence that the proprioceptively-perceived orientation of a freely-extended hand shows a very similar bias function (Li & Durgin, submitted), the present results seem to point to a stable and systematic bias in the perception of 3D orientation in reachable space.

7.1 A family of biases

• The authors description is intentionally limited to 3D slant in manual reaching space because it is fairly clear that the underfoot haptic perception of ramp orientation, for example, follows a rather steeper function than the one for manually reachable surfaces (Hajnal et al., 2011).
• The shape of that function has only been explored over a limited range, however.
• There seems to be some continuity between the nearspace bias function and biases shown in hill perception, once distance is taken into account.
• Durgin and Li (2011a) have argued that the 1.5 gain also applies to perceived gaze declination in the relevant range of declinations (i.e. out to about 45°).
• Second, in the present study, the bias function was very similar in shape when a very different set of numbers was required to produce it as a result www.intechopen.com Spatial Biases and the Haptic Experience of Surface Orientation 91 of labelling vertical as 0°.

7.2 Sine function scaling predicts the gain of 1.5

• The characteristic shape of the error functions the authors have observed somewhat resembles the first quarter cycle of a sine function.
• It appears fairly linear at the low end of the scale and compressive at the high end.
• It turns out that the sine function produces a bias function with a gain of essentially 1.5 at the low end of the scale.
• If the optical distance from the eye to a target is held fixed, then the sine function is proportional to the frontal vertical extent created between the www.intechopen.com Haptics Rendering and Applications 92 target and the horizontal plane at eye level.
• At present the authors only propose that the sine function appears to capture the shape of the 3D angular bias function remarkably well.

7.3 An implied direction of calibration?

• The idea that haptic scaling of perceived orientation during dynamic touch might be based on the ratio between the vertical extent of (finger-tip) travel and the total extent of travel is intriguing.
• Gravitational forces are highly relevant to haptics.
• The coding of orientation in this form would serve to express a useful ratio.
• When the vertical component equals the total extent of travel, the surface is vertical.
• An implied direction of calibration does not require that calibration always go in this direction, but only supposes that there is a natural basis for sinusoidal scaling of manual haptic slant perception and that this basis could then drive the visual scaling.

7.4 Scale expansion theory

• For large scale space, Durgin and Li (2011a) have proposed that the special role of proprioception of gaze direction in estimating distance (e.g., Wallach & O’Leary, 1982) may encourage scale expansion near horizontal with a gain of 1.5.
• Because their model provides impressive quantitative predictions of perceptual matching tasks (Li, Phillips & Durgin, 2011), it seems to capture an important feature of locomotor space perception.
• Durgin and Li have proposed that the 1.5 gain in the scaling of perceived slant may be driven by the 1.5 gain in gaze proprioception.
• That is, for horizontal ground surfaces to look flat requires a 1.5 gain in the optical slant.
• Thus, it might be argued that the expanded scale of perceived gaze declination also creates pressure for an expanded scale of visual slant.

8. Conclusion

• In this chapter the authors reviewed basic knowledge concerning spatial biases in the perception of slant and then presented novel experimental results.
• The authors experiments tested whether the perceived 3D orientation bias function for surfaces explored by dynamic touch was similar to that for visually perceived slant and static haptic touch.
• The authors have further suggested that the orientation bias function in manual reaching space resembles the first quarter cycle of a sine function.

Figures

Fig. 3. Verbal estimates of the (visual) slants of eight hills from Proffitt et al. (1995), ± 1 SEM. Hills were viewed from the base with gaze forward.

Fig. 8. The experimental set up. The orientation of the board could be rapidly adjusted to one of 16 orientations from 0° to 90°.

Full text

Swarthmore College Swarthmore College
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2012
Spatial Biases And The Haptic Experience Of Surface Orientation Spatial Biases And The Haptic Experience Of Surface Orientation
Frank H. Durgin
Swarthmore College
, fdurgin1@swarthmore.edu
Z. Li
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4
Spatial Biases and the Haptic
Experience of Surface Orientation
Frank H. Durgin and Zhi Li
Swarthmore College,
USA
1. Introduction
The two main purposes of this chapter are to review past evidence for a systematic spatial
bias in the perception of surface orientation (geographical slant), and to report two new
experiments documenting this bias in the manual haptic system. Orientation is a
fundamental perceptual property of surfaces that is relevant both for planning and
implementing actions. Geographical slant refers to the orientation (inclination or pitch along
its main axis) of a surface relative to the gravitationally-defined horizontal. It has long been
known that hills appear visually steeper than they are (e.g., Ross, 1974). Only recently has it
been documented that (1) there is also bias in the haptic perception of surface orientation
(Hajnal et al., 2011), and that (2) similar visual and haptic biases even exist for small surfaces
within reach (Durgin, Li & Hajnal, 2010).
To provide a context for understanding the present experiments, we will first provide an
overview of the prior experimental evidence concerning bias in the perception of geographical
slant. First we will discuss findings from both vision and haptic perception that have
documented perceptual bias for surfaces in reach. We will then review the literature on the
visual and haptic biases in the perception of the greographical slant of locomotor surfaces such
as hills and ramps. At the intersection of these two literatures is the historical use of haptic
measures of perceived geographical slant, and we will therefore review these measures with
particular attention to understanding some pitfalls in the use of haptic measures of perception.
We next contrast these haptic measures with proprioceptive measures of perceived orientation
and discuss the problem in interpreting calibrated actions as measures of perception.
Having laid out these various past findings we will then report two novel experiments that
demonstrate spatial biases in the haptic experience of real surfaces. The experiments include
both verbal and non-verbal methods modelled on similar findings we have reported in the
visual domain. Following the presentation of the experimental results we will discuss issues
of measurement in perception – especially pertaining to the interpretation of verbal reports,
and conclude with a discussion of functional theories of perceptual bias in the perception of
surface orientation.
2. Spatial bias in the perception of orientation: Surfaces within manual reach
What is meant by a spatial bias in the perception of surface orientation? Durgin, Li and
Hajnal (2010) reported a series of studies of a bias they called the “vertical tendency” in slant
www.intechopen.com

Haptics Rendering and Applications
76
perception. Specifically, they found that small, irregularly-shaped wooden surfaces
appeared steeper than they actually were both when viewed visually and when experienced
haptically while blindfolded. The term “vertical tendency” was used to distinguish the
observed effect from what has been called “frontal tendency” in the literature (Gibson,
1950). For many years it has been argued that surfaces viewed visually, appear compressed
along the depth axis of visual regard and thus appear more frontal to gaze than they are.
However, when Durgin, Li and Hajnal asked participants to make estimates of the
geographical slants of wooden surfaces within reach, they found that that they got
approximately the same bias function whether the surfaces were at eye level (so that
“frontal” and vertical coincided) or viewed with gaze declined by about 40°. Moreover, the
same kinds of bias were found when surfaces were experienced haptically by placing the
palm of the hand on them, though their measurements of this were limited to the angle of 0-
45°. The typical bias function for vision is shown in Figure 1.
Fig. 1. Surface orientation estimates for near visual surfaces presented within reach of the
hand (Durgin, Li & Hajnal, 2010, Experiment 1). Symbol size approximates SEM.
The estimates shown in Figure 1 are based on verbal/numeric estimates of orientation
relative to horizontal, but the bias observed cannot be due to verbal coding. Essentially the
same spatial function was found if participants instead estimated orientation relative to
vertical and their responses were then subtracted from 90° in order to express them relative
to horizontal. Thus for example, a surface that was actually at a 42° orientation from
horizontal (and thus 48° from vertical), was estimated as being about 60° from horizontal by
one group of participants and about 30° from vertical by another. Clearly both groups saw it
as much steeper than its actual slant. When the same 42° surface was explored, haptically,
by a third group of participants by each placing the palm of the right hand against it while
blindfolded, it was also judged to be 60° from horizontal (Durgin, Li & Hajnal, 2010,
Experiment 4).
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Spatial Biases and the Haptic Experience of Surface Orientation
77
Moreover, to emphasize that these biases did not depend on generating verbal estimates,
Durgin, Li and Hajnal (2010) asked a fourth set of participants to judge whether various
oriented planar surfaces were closer to horizontal or to vertical. They fit a psychometric
function to the resulting choice data and found that a surface slanted by only 34.3° from
horizontal was, on average, visually perceived to be equidistant from vertical and
horizontal.
This spatial bias function for near surfaces closely matches the observed proprioceptive
function for the perceived declination of gaze. That is, when people are asked to report the
pitch of their gaze, verbal reports provide evidence of an exaggerated deviation from
horizontal that closely matches the bias function shown above for perceived surface slant
(Durgin & Li, 2011a; Li & Durgin, 2009). Thus, it appears that several different perceptual
representations of pitch contain a bias that expands the scale of differences near horizontal
while compressing the scale near vertical.
Fig. 2. Surface orientation estimates for surfaces (0-48°) felt with the palm of the hand
(Durgin, Li & Hajnal, 2010, Experiment 4). Error bars indicate ±1 SEM.
In the haptic domain and in the proprioception of gaze, the perceptual scale of the bias
function for perceived pitch has mostly only been measured for orientations within about
50° from horizontal (e.g., Durgin, Li & Hajnal, 2010). In this range the scaling of pitch
tends to closely approximate a linear scale with a gain of 1.5 (Durgin & Li, 2011a). For
example the haptic data of Durgin, Li and Hajnal are shown in Figure 2. These data are
based on numeric estimates of orientation in deg (relative to horizontal) made based on
placing the palm of the hand on various real slanted surfaces while blindfolded. Durgin
and Li have reported a very similar function for explicit estimates of the pitch of gaze over
a similar range. Durgin and Li (like Durgin, Li & Hajnal) supplemented their verbal
estimation data with a horizontal-vertical bisection task and again found that a rather
shallow gaze declination of about 30° from horizontal was perceived as the bisection point
between vertical and horizontal gaze.
www.intechopen.com

Frequently asked questions

Q1. What are the contributions in "Spatial biases and the haptic experience of surface orientation" ?

In this paper, the authors reviewed the literature on the visual and haptic biases in the perception of the greographical slant of locomotor surfaces such as hills and ramps and reported two new experiments documenting this bias in the manual haptic system. 

Q2. How can the authors reduce the slant gain to a single free parameter?

By setting the slant gain to 1.5 and assuming an overall intercept of 0°, the authors can reduce the model to a single free parameter, based on the multiplier of log viewing distance. 

Q3. What is the guiding rule for haptic exploration of a surface?

The guiding rule might be that calibration occurs when there is some real possibility for action with immediate spatial feedback from more than one modality. 

Q4. What did they use to ensure that the steepest direction of inclination was felt?

In the present study the authors used real surfaces and provided a ridge along the main axis of the surface to ensure that the steepest direction of inclination was felt. 

Q5. How close is the perceived bisection point between horizontal and vertical?

In other words, across a variety of modalities (proprioception of gaze declination, visual perception of 3D surface orientation in depth and haptic perception of 3D surface orientation in depth) the perceived bisection point between horizontal and vertical is very close to 30° from horizontal. 

Q6. What is the effect of the special role of proprioception of gaze in estimating distance?

For large scale (locomotor) space, Durgin and Li (2011a) have proposed that the special role of proprioception of gaze direction in estimating distance (e.g., Wallach & O’Leary, 1982) may encourage scale expansion near horizontal with a gain of 1.5. 

Q7. How does the slant gain affect the perception of a hill?

Across a number of studies, Li and Durgin (2009, 2010; Durgin & Li, 2011a) have found systematic evidence suggesting that the visual perception of slant also has a gain of 1.5 in the low end of the geographical slant range. 

Q8. Why does the model provide such an impressive quantitative prediction of perceptual matching tasks?

Because their model provides impressive quantitative predictions of perceptual matching tasks (Li, Phillips & Durgin, 2011), it seems to capture an important feature of locomotor space perception.