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

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

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 striking coincidence or an impressive quantitative match.

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

Spatial Biases and the Haptic Experience of Surface Orientation 93

space is similar across visual and haptic modalities whether measured numerically with respect to vertical or with respect to horizontal or even when measured non-verbally using a horizontal-vertical bisection task. We have further suggested that the orientation bias

This research was supported by Award Number R15 EY021026-01 from the National Eye Institute. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Eye Institute or the National Institutes of Health.

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**9. Acknowledgments** 

**10. References** 

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target and the horizontal plane at eye level. It remains unclear why these specific ratios might be used as a scale of angle. It is possible that the sine function only incidentally matches the underlying bias function. Durgin and Li (2011a; see also Hajnal et al., 2011; Li & Durgin, 2009) have proposed that the shape of the bias function is driven by the utility of expanding the portion of the range used most frequently. At present we only propose that the sine function appears to capture the shape of the 3D angular bias function remarkably well.
