Thinking About Sampling

Now, suppose a particle moves around the circle at 9 revolutions per second, like this:

You might find the points on the circle moving so fast that they're hard to see clearly; it might also appear as if they're jumping around, sometimes seeming to rotate clockwise and other times counterclockwise. Essentially, this is because the screen has a certain refresh rate, and the computation for the animation also takes time, preventing us from presenting too high a speed of rotation.

So, we slow down the animation to ten times slower than real-time, effectively expanding the time axis by a factor of ten for observation, and we see the following result:

Now sampling ten times per second, since the particle makes 9/10 of a revolution every 1/10 second, we observe the following result. Please pay close attention to the blinking red dot; here, we present both the path of the particle and the sampling points at the same time for easy observation of their relationship:

Now, if we remove the original particle and only observe the changes in the sampling points, the result is as follows:

Clearly, if you could only see the sampling points, and record the time and position of their appearance, you might conclude that the particle is rotating clockwise around the circle, as shown below:

Exercise 2

1. Based solely on the recorded sampling points, what is the most direct assumption one might make about the particle's speed of rotation?
2. If both the particle's speed of rotation and the sampling rate are multiplied by a factor (for example, from a speed of 1 with 10 samples per second to a speed of 10 with 100 samples per second), what changes might be observed in the simulation?
3. What impact does increasing the particle's speed of rotation to an integer multiple of the sampling rate have, for example, sampling 10 times per second but increasing the speed from 1 to 11?
4. If you know the particle rotates at a constant speed, possibly clockwise or counterclockwise, but definitely not more than 10, is it possible to estimate the actual speed of rotation using the sampling record? If not, why? If so, what limitations does the sampling rate have?
5. If the particle's speed of rotation is exactly an integer multiple of the sampling rate, for example, a speed of 50 with a sampling rate of 10, what would the sampling result be?