I built a visualization to show how Fourier Series can “draw” any shape using rotating vectors[OC]

This project demonstrates the mathematical principle of the Fourier Series: the idea that any closed path can be represented as a superposition of rotating vectors (phasors).

I wanted to visualize exactly how adding more vectors improves the accuracy of the drawing, transforming abstract math into something you can see.

Fourier Series decomposes any periodic function into a linear combination of simple sine waves. In the complex plane, each sine wave corresponds to a rotating vector (or phasor).

This visualization shows:

• The Amplitude, Frequency, and Phase of each vector.
• How vectors connect "tip-to-tail."
• The path traced by the tip of the final vector.

What you are seeing (The 3 Levels) I created three demos to show increasing complexity:

1. The Circle: A single frequency (perfect demonstration).
2. The Heart: Medium complexity, showing how multiple frequencies coordinate to create corners/dips.
3. The Fourier Portrait: High complexity, demonstrating the massive expressive power of the series.

This is meant to be an educational tool. It helps bridge the gap between:

• Geometric interpretation of Fourier Series.
• Path drawing techniques in Computer Graphics.

Let me know if you have any questions about the Python implementation or the math behind it!