Dynamic Approximation of Static Quantities (Visual Depth Perception 14)
This Demonstration shows the asymptotic relation between and (binocular disparity) and between and (fixate convergence angle). It also shows the motion/pursuit law at the critical time where it is a better two-dimensional measure of depth than the time zero M/PL or the static formula (based on the symmetric case). The angle slider moves the distractor around a circle where retinal motion is constant. The "node percent" is the fraction of the eye radius where the node point lies with at the center of the eye.
For translation perpendicular to the line of sight at 6.5 cm/sec (interocular distance per unit time), retinal motion, the derivative of at , approximates binocular disparity provided the fixate is far from the eyes and the distraction is not near the eyes. (Specifically, they are asymptotic (retinal motion/binocular disparity ≈1) provided the distractor is not more than 45° nasal angle and you are not near the Vieth–Müller circle.) If the speed is changed by a factor , then is also multiplied by , so
, when (the approximation is good as a percentage of = B.D.).
The multiplier effect means that at high speed, the derivative is much larger than binocular disparity, so it can be detected more easily, even though, by itself, it is not sufficient to measure depth. (The rate of change would also be magnified, but and the motion/pursuit formula would be unchanged.) This means people can detect depth from motion when they cannot detect the depth of the same objects when at rest. For example, the retinal disparity of an object 20 m beyond a fixate at 100 m is min of arc, below what people can detect, while at , = 159 min/sec.
The retinal motion rate and motion/pursuit ratio at are constant on circles passing through the eye node and distractor with diameter on the axis. (These circles are similar to the invariant circles for binocular disparity, but slightly different.)
The pursuit rate satisfies and the convergence of the eyes on the fixate satisfies (when the eyes and fixate are symmetric). Sine and tangent are asymptotic to their argument for small angles, when ; for large fixate distances we may replace sine or tangent with the angle (and cosine with 1).
By simple geometry, the static relative depth for fixate and distractor on the axis is
The two asymptotic relations above then show that the static relative depth formula is asymptotic to the motion/pursuit law (at ):