The relative depth formula extends to the two-dimensional distractor as:
this is somewhat simpler:
This simplifies greatly at the time,
, when the eye crosses the
The motion/pursuit ratio
is constant on circles passing through the eye node at time zero and distractor
with diameter on the
axis. (These circles are similar to the invariant circles for binocular disparity, but slightly different.)
This means that the time zero motion/pursuit law is NOT an especially good indicator of the relative distance between the two-dimensional distractor and fixate,
, because you can move quite far from F on this circle with no change in the quantity
. If we take the translation of the observer into account, we can show that the peak value of the motion/pursuit law is a good indicator of the relative distance in two dimensions. (It is also likely that the changing value of the motion/pursuit ratio is a cue that the brain could use.)
The figure above on the left shows the peak values of the motion/pursuit law in color and the time zero M/PL in gray. On the right the dark gray graph is the two-dimensional relative depth.
, the motion pursuit ratio
all peak at
and this peak value satisfies
and the eye node is at the center of the eye,
, the motion/pursuit ratio and
have critical points at the times
The peak time motion/pursuit formula gives a value close to the relative distance
We do not know a theoretical explanation for the rough approximation of the peak motion/pursuit law by the 2D relative distance, but this Demonstration compares
, and the 2D signed relative distance,
. It also lets you vary time and compare the motion/pursuit law at other times. You can drag the distractor in two dimensions.
The Demonstration "Motion/Pursuit Law on Invariant Circles (Visual Depth Perception 4)" (see Related Links) lets you vary the distractor on the time zero invariant circles.