It is known that the perceived slants of large distal surfaces, such as hills, are exaggerated and that the exaggeration increases with distance. In a series of two experiments, we parametrically investigated the effect of viewing distance and slant on perceived slant using a high-fidelity virtual environment. An explicit numerical estimation method and an implicit aspect-ratio approach were separately used to assess the perceived optical slant of simulated large-scale surfaces with different slants and viewing distances while gaze direction was fixed. The results showed that perceived optical slant increased logarithmically with viewing distance and the increase was proportionally greater for shallow slants. At each viewing distance, perceived optical slant could be approximately fit by linear functions of actual slant that were parallel across distances. These linear functions demonstrated a fairly constant gain of about 1.5 and an intercept that increased logarithmically with distance. A comprehensive three-parameter model based on the present data provides a good fit to a number of previous empirical observations measured in real environments.

*gaze declination*(i.e., the perceived magnitude of the downward pitch of gaze), which was observed to be an exaggerated linear function of actual gaze declination, and perceived

*optical slant*(perceived surface orientation relative to the direction of gaze, Sedgwick, 1986). The model not only fit the empirical data of downhill slant perception quite well but also predicted an exaggerated linear function between perceived and actual optical slants (from 5° to 50°); this prediction of the model was confirmed later by direct measurement (Durgin & Li, submitted for publication; Durgin, Li, & Hajnal, 2010, Experiment 3).

*θ*

_{1}and the frontal extent BC subtends a visual angle of

*θ*

_{2}. The optical slant at ball A is

*β*.

*d*

_{1}is the frontal projected length of AB when gaze is directed at A.

*d*

_{2}is the frontal projected length of AB when gaze is directed at B. Because

*d*

_{1}equals to

*d*

_{2}cos

*θ*

_{1}, the aspect ratio (

*R*) between AB and BC (i.e., L/W) can be expressed as

*β*′, and actual optical slant,

*β,*can be expressed by

*d*

_{2}and

*W*are in the same fronto-parallel plane when gaze is directed to B, we assume that the perceived ratio between the two frontal extents is approximately equal to their actual ratio, i.e.,

*W*′ ≈

*d*

_{2}/

*W*(but see Higashiyama, 1992; it may be appropriate to correct for the vertical–horizontal illusion, VHI, which is typically measured as being about 1.05). Thus, Equation 3 can be simplified as

*θ*

_{1}(Foley, Ribeiro-Filho, & Da Silva, 2004; see also Murray, Boyaci, & Kersten, 2006), when

*θ*

_{1}is small, cos

*θ*

_{1}is still essentially 1.0. For example, even if a factor of 1.3 is assumed, when

*θ*

_{1}is less than 10°, cos

*θ*

_{1}would remain between 0.99 and 1.0. Thus, Equation 4 can be further simplified to

*β*′ can be deduced by measuring perceived aspect ratio

*R*′ in the aspect-ratio task. Whereas the aspect-ratio technique has been applied to the study of exocentric distances along the ground and interpreted in terms of geographical slant (e.g., Ooi et al., 2006), to our knowledge no published study has used it on slanted surfaces (but see Ooi & He, 2004). Although the trigonometry described by Equation 5 is similar to one of the equations published by Ooi et al. (in their Appendix A), the theoretical terms are entirely different. For example, our equation refers only to optical slant and includes no term for gaze declination; in contrast, the main equation of Ooi et al. includes terms for gaze declination and a term,

*η,*referring to an imputed constant additive bias in geographical slant perception; even when a term for optical slant is substituted for the gaze declination term, the geographical slant error term,

*η,*is retained. Thus, our equation was developed to express the perceived optical slant of any surface (e.g., a slanted surface floating in space), while the equation proposed by Ooi et al. concerns an imputed angular bias in the perception of a horizontal ground surface. These theoretical distinctions are important because our paper seeks to model perceived optical slant as a function of both distance and optical slant, rather than to propose a constant additive bias (

*η*) in geographical slant perception as Ooi et al. have done.

*β*

_{p}as a function of simulated slant

*β*and the log of viewing distance

*D*:

*k*

_{1}= 1.64 (95% CI: 1.57 to 1.72,

*t*= 44.4,

*p*< 0.0001),

*k*

_{2}= 6.96 (95% CI: 6.17 to 7.77,

*t*= 17.3,

*p*< 0.0001), and

*C*= −6.40 (95% CI: −9.93 to −2.91,

*t*= 3.07,

*p*= 0.0022).

^{1}

^{−8}) and the other started with a physical aspect ratio of 8.14 (i.e., 1.15

^{15}). On each trial, a two-alternative forced-choice (2AFC) response was collected by means of key presses to indicate whether the sagittal extent appeared longer or shorter than the frontal extent. The value (i.e., the physical aspect ratio) of the next trial in that staircase was adjusted up or down by a variable multiplicative step size, depending on the response given and the number of “turns” in that staircase so far (e.g., Durgin, 1995; a “turn” is defined when two consecutive responses to the same “staircase” series differ).

^{8}; this value declined to 1.15

^{4}after the first turn, to 1.15

^{2}after the second turn, and to 1.15 after the third turn, where it remained thereafter. Five turns for each of the 40 staircases were normally required to finish the experiment. Typically, about 300 trials were sufficient (which took about 25 min); the procedure also terminated if it reached 333 trials. The forty staircases were randomly interleaved with the relative probability of a staircase being selected on any given trial being proportional to the square of the number of “turns” remaining for that staircase. Thus, a staircase with 3 turns remaining was 9 times as likely to be selected as one with only 1 turn remaining. This rule served to roughly synchronize progress in the various staircases.

Physical ratio | 6° | 12° | 18° | 24° |
---|---|---|---|---|

1 m | 1.82 (0.10) | 1.80 (0.12) | 1.58 (0.07) | 1.45 (0.06) |

2 m | 2.27 (0.15) | 2.15 (0.12) | 1.79 (0.09) | 1.56 (0.06) |

4 m | 2.83 (0.19) | 2.42 (0.13) | 1.98 (0.08) | 1.68 (0.06) |

8 m | 3.30 (0.26) | 2.59 (0.13) | 2.14 (0.11) | 1.80 (0.08) |

16 m | 3.30 (0.28) | 2.75 (0.15) | 2.13 (0.10) | 1.76 (0.06) |

*k*

_{1}= 1.40 (95% CI: 1.34 to 1.46,

*t*= 47.8,

*p*< 0.0001),

*k*

_{2}= 2.88 (95% CI: 2.53 to 3.24,

*t*= 16.6,

*p*< 0.0001), and

*C*= 3.27 (95% CI: 1.67 to 4.81,

*t*= 2.93,

*p*= 0.0035),

^{2}i.e.,

*k*

_{1}= 1.49 (95% CI: 1.43 to 1.55,

*t*= 47.5,

*p*< 0.0001),

*k*

_{2}= 3.07 (95% CI: 2.67 to 3.42,

*t*= 16.5,

*p*< 0.0001), and

*C*= 3.28 (95% CI: 1.69 to 5.07,

*t*= 2.75,

*p*= 0.0063), i.e.,

^{3}Thus, the aspect-ratio task of Experiment 2 provides important confirmation that (1) the apparent dissociation in the effect of distance on the perceived slant ratio between Loomis and Philbeck (1999) and Proffitt et al. (1995) is clearly due to the different optical slant ranges that were tested. Moreover, (2) the aspect-ratio technique can be used as an implicit measure of perceived optical slant that provides a converging method of demonstrating this point. Because the aspect-ratio task may be susceptible to different forms of cognitive correction (e.g., Granrud, 2009) than the verbal slant estimation data, we regard the quantitative divergence between the two tasks as less important than the qualitative convergence.

*η*) in geographical slant perception, which ought to imply a tilted planar surface.

*integrated*into an increasingly elevated ground plane. Our model is currently silent on exactly how perceived distance should be understood along each theoretical ray of sight from the eye to the ground. In this sense, our model should be interpreted as accepting that there is a dissociation between perceived location (along the ground) and perceived slant (at each point). This distinction between location and shape has a long tradition (e.g., Loomis & Philbeck, 1999). Our theory captures the distinction in terms of differences between these two separate angular variables, gaze declination and optical slant. Optical slant is primarily a visual variable, the perception of which evidently varies with viewing distance, whereas gaze declination is primarily a proprioceptive variable, which is probably not so affected by viewing distance (but see O'Shea & Ross, 2007). Thus, evidence of dissociations between perceived egocentric distance and perceived exocentric distance as a function of distance (e.g., Loomis et al., 1992) is consistent with our theory (for an alternative view, see Wu, He, & Ooi, 2008).

*η*. However, their published estimates of

*η*have varied from about 3° (Ooi & He) to about 14° (B. Wu et al.). Thus, their model has a free parameter (

*η*) that has varied by a factor of five from one context to another. Despite this, errors of even 14° are simply not large enough to account for the much larger errors in geographical slant perception reported by Proffitt et al. (1995), nor do they capture the effects of viewing distance that have been documented by Bridgeman and Hoover (2008). In contrast, our studies have consistently found evidence for an angular scaling factor of about 1.5 in perceived optical slant (Durgin & Li, submitted for publication; Durgin, Li et al., 2010; Li & Durgin, 2009).

*χ*

^{2}(1) = 1.03,

*p*= 0.3092, whereas one that included only a linear term provided a reliably worse fit to the data than one that also included a logarithmic term,

*χ*

^{2}(1) = 27.4,

*p*< 0.0001. Thus, these data strongly support the use of a logarithmic distance term. The model gain parameter for slope for distances of 1–8 m was 1.68 (95% CI: 1.60–1.76).

*χ*

^{2}(1) = 1.18,

*p*= 0.2756, whereas one that included only a linear term provided a reliably worse fit to the data than one that also included a logarithmic term,

*χ*

^{2}(1) = 29.1,

*p*< 0.0001. Thus, these data, like those of Experiment 1, strongly support the use of a logarithmic distance term in the model. The logarithmic distance model slant gain parameter for distances of 1–8 m was still 1.40 (95% CI: 1.34–1.46).