Fourier Analysis: Drawing Llamas with Circles
An educational guide explains how to use Fourier transforms to draw complex shapes like llamas using nested rotating circles.
read it The piece connects abstract Fourier series theory directly to practical JavaScript code for rendering arbitrary shapes.
summary
The article explains how any closed continuous curve can be approximated using a series of rotating vectors driven by Fourier analysis. It walks through the underlying mathematics, covering complex numbers, linear algebra, and Euler's identity via a square wave example. Finally, it demonstrates how to sample an SVG path of a llama and compute its coefficients using a Discrete Fourier Transform in JavaScript.
key points
- Chaining rotating circles with specific frequencies, radii, and phases can reproduce any closed curve.
- The Fourier transform maps time-domain signals to frequency-domain basis functions using inner products and complex exponentials.
- A basic Discrete Fourier Transform implemented in JavaScript extracts these coefficients from sampled SVG path coordinates.
who should read it
Software engineers and math enthusiasts interested in signal processing and creative visualization.
discussion warm
consensus
Commenters express delight at the visual math application, sharing alternative projects and recommending complementary video explanations.
pushback
A commenter critiques a confusing explanation of velocity and position derivatives in the complex exponentials section.
I don't know if you're familiar with the 3blue1brown YouTube Channel, but he does an outrageously good job of explaining this mechanism in about 20 minutes. Search YouTube for "3b1b de4" and it should be the first video: "But what is a Fourier series? From heat flow to drawing with circles".
It directs readers to a widely praised video that provides an intuitive visual breakdown of Fourier drawing mechanics.
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