Fourier Epicycles Simulator

The core of the simulator is the Discrete Fourier Transform (DFT). It analyzes a set of N sampled points from your drawing (treating the x and y coordinates as separate complex numbers) and outputs a set of complex coefficients, one for each frequency component.

Here, $x[n]$ is the n-th sampled point of your drawing (as a complex number: $x + iy$). $X[k]$ is the complex Fourier coefficient for frequency $k$. The magnitude $|X[k]|$ is the radius of the k-th circle, and the argument $\arg(X[k])$ is its starting phase (angle).

The reconstruction, which you see as the epicycle animation, is the Inverse DFT. Each term describes a vector rotating at a specific frequency. The sum of all these rotating vectors traces your original path.

In the animation, each term $X[k] e^{2\pi i k t / T}$ represents one epicycle. Its angular velocity is $\omega_k = 2\pi k / T$. The final "pen" position is the vector sum of all these rotating arms. When you adjust the "Circles N" slider, you're changing how many terms of this sum are used in the approximation.

Signal Compression (JPEG/MP3): Fourier analysis is the backbone of modern compression. For instance, an MP3 encoder uses a similar transform to break down audio into frequency components, then discards the ones humans are less likely to hear, drastically reducing file size.

Medical Imaging (MRI): Magnetic Resonance Imaging machines don't take a direct picture. They measure the Fourier transform of the spin of atoms in your body. The complex image is then reconstructed using the inverse Fourier transform, just like this simulator rebuilds your drawing.

Radio & Communications: Every Wi-Fi or Bluetooth signal relies on manipulating different frequency components. By assigning data to specific Fourier components (a method called OFDM), multiple devices can share the same radio channel without interfering.

Vibration & Structural Analysis: Engineers use Fourier transforms to diagnose problems in machinery. For example, by measuring vibrations from a car engine and decomposing them into frequencies, they can pinpoint which rotating part (like a specific bearing) is wearing out.

When you start using this simulator, many people encounter similar questions or mistakes. Let me explain three key points you should grasp first.

1. "Does maximizing the number of circles N guarantee perfection?"
It's true that in theory, using as many circles as the original number of points allows for perfect reproduction. However, the "number of points" itself is finite within the simulator; even the smooth curve you draw with your mouse is captured as a collection of discrete points. Therefore, even with N maximized, the reproduction is merely "passing through discrete points." In practice, finding the "minimum N that achieves the required accuracy" is more important than perfect reproduction. Think of it like MP3 compression cutting inaudible high frequencies above 20kHz.

2. "Can the rotation speed of each circle be chosen freely?"
While you change the "speed" with the slider, it's easy to misunderstand: the ratio of relative rotational speeds between circles is fixed. The angular velocity of the k-th circle is determined by $\omega_k = \frac{2\pi k}{T}$, making it an integer multiple of the fundamental speed. Consider what you're changing in the tool as the overall "playback speed," which uniformly speeds up or slows down this fundamental speed. To change frequencies in actual design, you need to alter the number or spacing of the sampling points themselves.

3. "Is the phase (initial angle) just decorative?"
Though less flashy than radius or speed, phase is a critically important parameter that determines the shape. For example, two circles with the same radius will trace completely different elliptical paths for the pen tip depending on whether their phases are offset by 90 degrees. When reconstructing a signal from Fourier coefficients in practice, incorrect phase information badly distorts the waveform. In this simulator too, you can see that each circle's arm starts pointing in a different direction from the very beginning.

Fourier Epicycles Simulator is a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.

By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition — making it an effective learning tool for students and a rapid-verification tool for practicing engineers.