We recently discovered a new formula for visual depth perception from motion parallax. Neurological work by J. W. Nadler, D. E. Angelaki, and G. C. DeAngelis in "A Neural Representation of Depth from Motion Parallax in Macaque Visual Cortex" (

*Nature*,

**452**(7187), 2008 pp. 642–645) suggests that an extra-retinal signal is needed for depth perception. In the static case, retinal disparity and motor control convergence are both needed to mathematically determine depth. Our dynamic formula involves a moving retinal image cue and an eye pursuit motor control cue that we believe is the needed extra-retinal signal.

Mark Nawrot conducted psychophysical experiments that indicate people use the motion/pursuit ratio to determine depth from motion. (Our joint work has not yet appeared.) Previous work by Nawrot and others (such as M. Nawrot and L. Joyce,

*Vision Research*, "The Pursuit Theory of Motion Parallax,"

**46**(28), 2006 pp. 4709–4725) suggested that the motion/pursuit ratio was important. Our new formula forms a theoretical basis to understand past work, suggest new experiments, and perhaps even find a neurological basis for the formula.

*Mathematica* 6 played a role in this discovery and is helpful in explaining both the new dynamic formula and the old static formula for depth. There is a remarkable relation between the dynamic and static formulas. This collection of Demonstrations explains the new and old formulas interactively and lets you make your own computations. Perhaps the computations will be helpful in designing new experiments.

We begin with a description of the formula for depth from motion parallax in a symmetric case with links to Demonstrations that explain basic terms and quantities. We outline the other visual depth perception Demonstrations below.

The derivative in terms of the

eye parameters, the node percent

, interocular distance

, and eye radius

above is

.

This peaks at

(when the denominator is largest):

.

The observer's translation also causes the

angle separating the fixate and distraction to change, causing motion of the image of

on the retina. In this case, where the distractor is also on the

axis, at

this derivative is

.

The ratio of retinal motion over pursuit is

.

The basic case of the motion/pursuit law for relative depth from motion parallax is

where the angle

simplifies the math, but may not correspond to a neurological signal.

**The Visual Depth Perception Demonstrations**