Fourier Series Visualization

Draw any shape and watch it be recreated with a sum of rotating vectors (epicycles).

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About This Simulation

This interactive animation demonstrates the concept of the Fourier series. The core idea, discovered by Joseph Fourier, is that any periodic signal (or in our case, a closed 2D shape) can be represented as a sum of simple sinusoids (sines and cosines). In the complex plane, this is visualized as a series of rotating vectors, often called epicycles.

Each vector represents a term in the Fourier series. The length of the vector corresponds to the amplitude of the sinusoid, and its rotation speed corresponds to the frequency. By adding these vectors tip-to-tail, the final point traces out the original shape.

How to Use

Future Directions

This visualization could be extended in several ways: