You can create your own compelx Fourier Series visualizations as shown here. You can make anything you want really, I originally got this idea from 3Blue1Brown's video here . Mathologer also casts this result as a question of paths traced by epicycles, the video can be found here. For some cool examples see the examples section.

1. Examples

Here are some examples with the corresponding code to generate them.

python fourier_visualization.py path/to/img .05 1 1 0

python fourier_visualization.py path/to/img .2 1 1 1

python fourier_visualization.py path/to/img .1 1 0 1

2. Usage

To use this script simply open a terminal window, navigate to the root directory where the code is located. For example,

cd /path/to/root/

Then run the following line which takes in four arguments

python fourier_visualization.py path/to/img frac_points_used export_animation use_main_curve sort_by_amplitude

frac_points_used is a value between 0 and 1 and specifies the number of points to use for the final image, the more points the more detailed the the slower it becomes. If the string None is passed, all points are used. export_animation is a bool and is therefore either 0 or 1, 1 for export animation. use_main_curve produces an image using only the main curve found in the image, for more complicated images like the titanic, this should be set to 0. sort_by_amplitude sorts the phasors in ascending order based on their amplitudes, otherwise they are sorted based on ascending frquency (recall $\pi$ is the maximum angular frequency).

3. Theory

The rich theory of Fourier Analysis has become one of the most utilized fields by the applied sciences. The theory is applicable to many classes of functions. Here we consider the complex Fourier Series which possess a quick numerical implementation named the FFT. A Fourier Series is the expansion of a sufficiently well-behaved function in a basis set of complex exponentials of varying frequencies. That is

$$ f(n) =\frac{1}{N} \sum_{k=0}^{N-1} c_k \exp(j2\pi k n / N) $$

where N is the period of the signal in samples. The coefficients $c_k$ encode the "strength" of each frequency in the signal. If we look at this expansion in the complex plane then these are simply phasors tip-to-tip which are the circles the circles you can see in the demonstration above. The radius of each circle is given by $|c_k|$ and the phase for a given n is given by the complex exponential offset by the phase of $c_k$.