Left and right eyes

We now address how the transformation chain (3.41) is altered for stereoscopic viewing. Let $ t$ denote the distance between the left and right eyes. Its value in the real world varies across people, and its average is around $ t = 0.064$ meters. To handle the left eye view, we need to simply shift the cyclopean (center) eye horizontally to the left. Recall from Section 3.4 that the inverse actually gets applied. The models need to be shifted to the right. Therefore, let

$\displaystyle T_{left} = \begin{bmatrix}1 & 0 & 0 & \frac{t}{2}  0 & 1 & 0 & 0  0 & 0 & 1 & 0  0 & 0 & 0 & 1  \end{bmatrix} ,$ (3.50)

which corresponds to a right shift of the models, when viewed from the eye. This transform is placed after $ T_{eye}$ to adjust its output. The appropriate modification to (3.41) is:

$\displaystyle T = T_{vp} T_{can} T_{left} T_{eye} T_{rb} .$ (3.51)

By symmetry, the right eye is similarly handled by replacing $ T_{left}$ in (3.51) with

$\displaystyle T_{right} = \begin{bmatrix}1 & 0 & 0 & -\frac{t}{2}  0 & 1 & 0 & 0  0 & 0 & 1 & 0  0 & 0 & 0 & 1  \end{bmatrix} .$ (3.52)

This concludes the explanation of the entire chain of transformations to place and move models in the virtual world and then have them appear in the right place on a display. After reading Chapter 4, it will become clear that one final transformation may be needed after the entire chain has been applied. This is done to compensate for nonlinear optical distortions that occur due to wide-angle lenses in VR headsets.

Steven M LaValle 2016-12-31