**9. Acknowledgments**

92 Haptics Rendering and Applications

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

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 (fingertip) 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 there is no vertical movement, the surface is horizontal. When the vertical component equals the total extent of travel, the surface is vertical. When the vertical extent is half the total extent of travel (i.e., sin(30°) = 0.5), the surface is "half-way" between horizontal and vertical (i.e., the perceived bisection point). These facts fit our data well and seem appropriate for manual haptics. It is much less clear why the visual perception of the slant of manually-reachable surfaces should be similarly coded, given that neither the vertical extent of orientation change nor the overall surface extent are directly given in vision. However, if vision is calibrated to haptic experience, this would be sufficient to suggest why the visual and haptic codings are aligned, but also why the haptic experience of individuals who are congenitally blind are distorted in a similar manner. Insofar as it depends on manual haptic exploration, this hypothesis would apply specifically to manually-reachable surfaces. 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

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. 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.

In this chapter we reviewed basic knowledge concerning spatial biases in the perception of slant and then presented novel experimental results. Our 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. We found evidence that supports the view that the spatial bias for the perceived 3D orientation of surfaces in manual reaching

well.

**7.3 An implied direction of calibration?** 

this basis could then drive the visual scaling.

**7.4 Scale expansion theory** 

**8. Conclusion** 

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. This work was also supported by a Swarthmore College Faculty Research Grant.
