Problem 5.15

Nasser Abbasi

Wheel appears to turn back due to temporal aliasing. This happens when the wheel spokes are rotating at frequency higher than the nyquist frequency. Nyquist frequency is 1/(2*sample interval). The sample rate here is the interval between each picture frame. When the wheel spokes rotate at a frequency higher than nyquest frequency, then the actual frequency seen is the alias frequency and not the real actual frequency the wheel spokes are rotating at.

To explain this, I assume a wheel with 4 symmetrical spokes (i.e. 90 degrees angle between each spoke and the other). Assume the wheel in real life is spinning in the clock wise direction.

If the wheel is rotating 90 degrees per picture frame, then the wheel will appear to be stationary (i.e at each frame, the wheel looks exactly the same as it did in the previous frame). I.e. assume the sample interval is such that it is equal to the time it takes for the wheel to make ¼ revolution.

Now, assume the wheel is rotating at 89 degrees per the time it takes to take one frame. So at each frame each spoke moves 89 degrees clock wise, but this is the same as if the spoke moved 1 degree anti clock wise. So, the eye sees the smaller degree movement as the actual movement and see the wheel actually moving at a rate of 1 degree per frame but in the anti clock wise direction (opposite from its actual movement in real life).

The same for 88 degrees, the eye will see the wheel moving 2 degrees per frame in the anti-clock wise direction. This process will continue. When the wheel rotates at 46 degrees in the clock wise direction, it will appear to rotate 44 degrees in the opposite direction. At 45 degrees rotation per frame, one will not know which direction it is rotating. When the wheel rotates at 44 degrees per frame in the clock wise direction, then it will appear to rotate in the forward direction which matches the actual real life movement.

So, for a 4 sopke wheel, we can’t turn the wheel by more than 45 degrees per frame (or 1/8 revolutions per frame) without seeing this temporal aliasing.