Draw a line.
Watch it become circles.

Every closed doodle can be rebuilt from rotating circles. Draw your own shape, or pick a specimen from the menagerie. Then watch the Fourier series find its way home.

256 samples · real mathematics

01 / the original

Your continuous scribble

Draw here with a mouse
or your finger ↘

Try drawing a shape in one stroke · it closes automatically

Or borrow a shapeOne stroke, infinitely many harmonics

02 / the reconstruction

An orchestra of rotating circles

12 active harmonics

● tracing nowphase · 0%

What is happening?The bright point is the sum of all spinning vectors.

A large circle makes a broad sweep. Smaller circles add finer details. Even sharp corners eventually emerge from the accumulated motion.

Number of circles12 / 128

1 · simple128 · intricate

Try 1 → 4 → 12 → 40. The curve sharpens as you add detail.

Control the motionspace to pause

Animation speed1.0×

Circles Full curve

The invisible arithmeticParseval's identity

Sample energy captured

—

of original curve

RMS sample error

—

in drawing units

Why does this work?

The drawing is sampled at 256 equally spaced positions along its length, with the last point joined back to the first. Treat each position as a complex number z = x + iy. The discrete Fourier transform expresses those samples as rotating vectors:

z(t) ≈ ∑_k c_k e^2πikt

Each coefficient c_k supplies one circle's radius and initial angle; k gives its signed rotation speed. The slider keeps the largest coefficients. Parseval's identity makes the displayed sample error exact for that selection: omitted squared coefficient magnitudes add up to the squared reconstruction error at the sample points.

The frequencies inside your drawing

largest coefficients first ↗

Each bar is a rotating circle. The glowing bars are included; faint bars are omitted. A complicated silhouette is often surprisingly economical.

Built from scratch · finite discrete Fourier transform · no servers, trackers, or libraries required