I've completed my graduation in Computer Engineering and I've used Fourier Transform many times, yet when I see this animation, I fail to understand how they're realized.
Can anyone please explain me how the angular velocities of each circles in every diagram is related to each circle of the same diagram of the given equation?
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You start with some function $f$ defined on the interval $[-\pi, \pi]$ that has the property $f(x) = -f(-x)$ (in this case, some rectangular signal was used). This function can be represented as $\sum_{k=1}^\infty c_k \sin(k x)$ with some coefficients $c_k$. Using either some calculus or a numeric computation, you compute $c_k$. Then you simply draw a bunch of circles, with $j$-th circle having the center at position $\sum_{k=1}^j c_k [\cos(k x), \sin(k x)]$ and radius $c_j$. You also draw a point at $\sum_{k=1}^{j_{max}} c_k [\cos(k x), \sin(k x)]$. Obviously, the vertical coordinate of this point is $\sum_{k=1}^{j_{max}} c_k \sin(k x) \approx f(x)$, so you can draw a horizontal line at this height to emphasize this relationship.
Cute visualization, by the way...
As the text in the animation tries to indicate, the radii are in proportions $1:\frac13:\frac15:\frac17$ and the angular velocities $1:3:5:7$.