The horizontal disparity pattern associated with slant about a vertical axis can be represented locally as a horizontal gradient of horizontal disparity or, alternatively, as a horizontal size ratio (
HSR), which is the ratio of horizontal angles a surface patch subtends at the left and right eyes (
Rogers & Bradshaw, 1993). Changes in
HSR cause obvious changes in perceived slant, but
HSR by itself is an ambiguous slant indicator because it is also affected by the plane’s position relative to the head (
Backus, Banks, van Ee, & Crowell, 1999;
Gillam & Lawergren, 1983;
Ogle, 1950). Thus, to estimate slant about a vertical axis, the visual system employs other signals to aid the interpretation of horizontal disparity. These signals include vertical disparity (which can be quantified by the vertical size ratio [
VSR]), eye-position signals (indicating the horizontal version and vergence), and other slant signals, such as the texture gradient (
Backus et al, 1999). The horizontal disparity pattern associated with slant about a horizontal axis (right panel of
Figure 1) can be represented locally as a horizontal-shear disparity.
Ogle and Ellerbrock (1946) defined this disparity as follows. A line through the fixation point and perpendicular to the visual plane is a vertical line. There is a horizontal axis through the fixation point, in the visual plane, and parallel to the interocular axis. We rotate the vertical line about this axis and project the images of the line onto the two eyes. The horizontal-shear disparity (
HR) is the angle between the projections of the line in the two eyes. If the eyes are torsionally aligned (ie, the horizontal meridians of the eyes are coplanar) and fixating in the head’s median plane, slant about a horizontal axis is given by:
where
S is the slant,
i is the interocular distance, and
d is the distance to the vertical line’s midpoint. When the distance to the surface is much greater than the interocular distance, slant is given to close approximation by:
where
μ is the eyes’ horizontal vergence (right panel,
Figure 1).
1,
2 Thus, estimating slant about a horizontal axis is straightforward when the eyes are aligned: the visual system must only measure the pattern of horizontal disparity (quantified by
HR) and the vergence distance (
μ), which could also be measured by use of the pattern of vertical disparities (
Rogers & Bradshaw, 1995;
Backus et al, 1999).