ABC Tools
Fourier Epicycles illustration

Fourier Epicycles

Any closed path can be rebuilt from rotating circles (epicycles) whose radii and speeds come from the discrete Fourier transform of the path. Draw a closed curve with the mouse, or pick a preset shape — a star, a heart or the letter π — and the tool computes its DFT and animates a chain of circles, each turning at an integer frequency, whose tip traces out the reconstruction next to your original drawing. A slider sets how many terms (circles) to use, so you can watch a handful of circles already capture the rough shape and each extra one add finer detail. Play, pause, clear and redraw.

Runs 100% in your browser — simulations are computed locally on your device.

Read the full guide to this tool

Notes

  • The path is treated as a list of complex numbers; the discrete Fourier transform turns it into circles with a radius, phase and integer rotation speed each.
  • Circles are drawn largest first; adding terms improves the fit, and with as many terms as points the trace is exact.
  • Sharp corners need many high-frequency terms — a few low frequencies already give a smooth approximation of the overall shape.
  • Runs 100% in your browser — simulations are computed locally on your device.

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