**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. Although numerically lower than the 34° average reported for visual slant by Durgin, Li and Hajnal (2010), this difference was not statistically reliable. The subjective bisection point did not differ reliably from 30°, *t*(11) < 1, but did differ reliably from 45°, *t*(11) = 6.89, *p* < .0001.

Spatial Biases and the Haptic Experience of Surface Orientation 91

of labelling vertical as 0°. Finally, the third important fact that argues against a purely numeric bias is that a very different bias function emerges when 2D orientation (of lines on a plane) is studied using similar numeric methods (Durgin and Li, 2011b). In this case, there is still a bias function that exaggerates deviations from horizontal, but absolute signed error

The characteristic shape of the error functions we have observed somewhat resembles the first quarter cycle of a sine function. Such a function is plotted in Figure 11, scaled to 90°. Here we will first consider the features of the sine function that render it a promising model.

Fig. 11. Sine of slant, scaled to 90°. The sine function has a gain of essentially 1.5 over the

There are two features of particular note in Figure 11. First, the sine function captures the main shape of the bias functions we have been discussing: It appears fairly linear at the low end of the scale and compressive at the high end. The second point is made graphically by the line in Figure 11 representing a gain of 1.5 from horizontal. 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. Given the variety of empirically observed angular bias functions that have proven to have nearly exactly that gain near horizontal (e.g., visually perceived optical slant, haptically and visually perceived geographical slant, perceived gaze declination), this seems like either a

A sine function represents the ratio between the surface length and the vertical extent of the surface. Unlike grade, which corresponds to the tangent function (rise over run), the sine function would seem to place priority on surface length, which has the virtue of being an ecologically relevant variable. For perceived gaze declination along a ground plane, where the relevant vertical extent is normally the observer's eye height, the sine of gaze declination corresponds to the reciprocal of the optical distance from the observer's eye to the point at the center of regard along the ground. 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

range from 0 to nearly 45°. Means of pooled visual and haptic data are shown.

striking coincidence or an impressive quantitative match.

peaks at 30° rather than at 60° as for the 3D function.

**7.2 Sine function scaling predicts the gain of 1.5** 
