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

0.372 min of arc, below what people can detect, while at

100 km/hr,

= 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

):

.