On the Perception of Moving Contours by a Single Eye

G.R.Dixon, 5/20/2006

1. Introduction.

In this article we compute what a single "eye" would hypothetically see when certain lines/contours move. It is not a trivial matter, since at time t=0 the eye is acted upon by photons that depart from points on the line/contour at different moments in the past (and not by photons that leave these points at a common instant). Four lines/contours are investigated: (1) a horizontal line moving in the positive x-direction; (2) a vertical line moving in the positive x-direction; (3) a circular contour moving in the positive x-direction; and (4) a clockwise-rotating straight line which, at t=0, coincides with the x-axis.

The observing eye is permanently at rest at point z on the positive z-axis, where z is comparable to a given line’s/contour’s rest dimensions. All of the lines/contours are assumed to lie in the xy-plane. In the cases of the translating lines/contours, the line’s/contour’s center moves along the x-axis at the constant, relativistic speed of .95c. At time t=0 these lines’/contours’ centers are at the origin. In the case of the rotating line, the line center is permanently at rest at the origin and its ends move at a speed of .95c.

The general approach is to break each line/contour up into a large number of segments, with each segment approximating a point source of photons. Photons are presumably emitted constantly and in all directions from each such point. However only one photon, emitted by a given point at the retarded time, can reach the eye at time t=0. This photon will presumably determine where the eye __perceives__ that particular point to be at time t=0. (It will, of course, not be where the point actually __is__ at time t=0.) Collectively all of the photons, reaching the eye at time t=0, determine the line’s/contour’s __perceived__ shape.

2. Horizontal Line Segment Translating Along the x-axis.

Let us assume that a straight line segment, of rest length L = 2 meters, moves in the positive x-direction at constant speed .95c. The segment is coincident with the x-axis. At time t=0 the segment’s center is at the origin. Theoretically its __measured__ length would be length-contracted to g^{-1}L. The question is, what would the segment __look__ like to the eye at z=L? Fig. 2_1 depicts the retarded points (i.e. the __apparent__ segment). At t=0 the entire segment appears not yet to have reached the origin. Furthermore, the apparent length is more than 3 times the rest length (and many more times the contracted length)!

Figure 2_1

Apparent Horizontal Line Segment

3. Vertical Line Segment Translating Along the x-axis.

Fig. 3_1 depicts the line segment of Sect. 2, rotated 90 degrees (all other things being equal). At time t=0 the middle of the segment is perceived to be well to the left of the origin, and points above/below the middle are perceived to be even further to the left.

Figure 3_1

Apparent Vertical Line Segment

4. Contour is a Circle (when resting).

Fig. 4_1 depicts the contour that is a circle of radius R = 1 meter in its rest frame. When moving at a speed of .95c the circle would theoretically be __measured__ to be length-contracted into an oval. However, this is not what is __perceived__. Note that the observing eye in this case has been located further out on the z-axis, at z = 10R.

Figure 4_1

Apparent Circular Contour

5. Rotating Line Segment.

Figure 5_1 depicts the eye’s perception of a straight line, of length L=2 meters, that rotates clockwise. At time t=0 the line coincides with the x-axis. The observing eye is at z=L/2.

Figure 5_1

Apparent Rotating Line Segment