Fourier Epicycles: Interactive Visualization

Fourier epicycles in this visualization

Walk once round a closed curve in the plane and read each point as a complex number $z(t)$ , with $t$ from 0 to 1 for one turn. Its complex Fourier series writes $z(t)$ as a sum of terms $c_k e^{2\pi i k t}$ , one for each whole number $k$ . Each term is a point going round a circle of radius $|c_k|$ , $k$ times per turn, starting at the angle $\arg c_k$ ; negative $k$ turn the other way. Put the circles tip to tail and the last tip draws the sum. $c_0$ does not turn: it is where the chain starts.

z(t) = \sum_{k} c_k\, e^{2\pi i k t},\qquad c_k\, e^{2\pi i k t}:\ \text{radius } |c_k|,\ k \text{ turns per turn, starting at } \arg c_k

What the page computes

The page samples the curve at $N = 1024$ points, evenly along its length (or, for the heart, evenly in its formula’s $t$ ), and computes the discrete coefficients $\tilde c_k$ . All $N$ terms together pass exactly through the $N$ samples. The error at the samples is $\sqrt{\sum_\text{unused} |\tilde c_k|^2 / \sum |\tilde c_k|^2}$ (without $c_0$ ): by the discrete Parseval identity it is the root-mean-square distance between the drawn curve and the samples, relative to the shape’s size. It is not a comparison with your original stroke.

Build up adds circles level by level, $M = 1, 2, 4, 8, \ldots$ , one turn each, largest first (or lowest frequency first). Every finished level stays as a faint ghost: where the levels agree the light piles up, where they still disagree it spreads. One circle is a circle; $k = 1$ and $k = -1$ together make an ellipse with semi-axes $|c_1| + |c_{-1}|$ and $\bigl||c_1| - |c_{-1}|\bigr|$ .

The ticks on the pen’s trace are $1/64$ of a turn apart in time. Where they crowd, the pen went slowly over that stretch on average; between two ticks a fast circle can turn several times, so they do not show the speed at an instant.

Easy to misread

The coefficients belong to the curve and to the way it is walked. The heart $x = 16\sin^3 t$ , $y = 13\cos t - 5\cos 2t - 2\cos 3t - \cos 4t$ walked by its own $t$ has exactly 8 non-zero terms; walked at constant speed, the same heart needs 21 circles to get under 1 % error.
The order of the chain changes how it looks, not where its tip goes: a sum does not depend on the order of its terms.
At corners, as in the square or the star, the partial sums can stick out beyond the shape while there are few circles, but this overshoot dies away as circles are added (not with every single one). Jumps behave differently: there the overshoot lasts (the Gibbs phenomenon). Because this page uses $N$ samples, a small wobble between the samples remains even with every term in: about 0.0007 for the square of half-side 1.
The glow, the trails and the fading of the circles are display only.

History

Ancient astronomers described the planets with circles riding on circles, the deferent and the epicycle. A chain of uniformly turning circles is exactly a sum of terms $c_k e^{2\pi i k t}$ , which is why enough epicycles can follow any closed path [2]. Fourier introduced his series in 1807 for the flow of heat [1].

Related concepts

Complex plane · Unit circle · Fourier series (page planned) · Parametric equation (page planned)

References

J. Fourier, Théorie analytique de la chaleur, Firmin Didot, Paris (1822); the 1807 memoir to the Institut de France.
N. R. Hanson, “The mathematical power of epicyclical astronomy”, Isis 51 (1960) 150–158.

Further reading: Wikipedia: Fourier series; MacTutor History of Mathematics: Joseph Fourier.