Draw any shape with rotating circles - the magic of complex Fourier series
Chain enough spinning circles together, tip to tail, and the last one can trace any closed curve you like, including one you draw yourself.
Draw a closed shape on the canvas, or pick a preset, and watch the circles redraw it. Drag the Circles slider down to a handful, then back up, and see the detail come and go.
Draw on the canvas to create your own shape, or choose a preset. Watch as rotating circles (epicycles) recreate your drawing through the magic of Fourier series!
One of the most beautiful applications of Fourier analysis is the ability to recreate any closed curve using nothing but rotating circles. Each circle (or epicycle) rotates at a different frequency, and when you trace the tip of the final circle, the path draws your shape.
This works because the complex Fourier series decomposes your drawing into circular motion: each term c_n e^(inωt) represents a circle rotating at frequency n with radius |c_n|.
Given a closed path f(t) where t goes from 0 to 1, we compute the complex Fourier coefficients:
Each coefficient c_n tells us about a circle rotating at frequency n:
The key insight is that circular motion is the most fundamental form of periodic motion. Any periodic function can be built from circular motions at different frequencies:
Few Circles: Captures the general shape but misses fine details. Like a blurry photo.
Many Circles: Captures sharp corners and fine details. More circles = higher resolution.
The word "epicycle" comes from ancient astronomy. Greek astronomers like Ptolemy used epicycles (circles upon circles) to model planetary motion. While their geocentric model was wrong, the mathematics they developed is essentially the same as the complex Fourier series - any periodic motion can be decomposed into circular components!
Spot a mistake or have an idea?