Fourier analysis: writing with waves
Back to visual explanationsTwo signals. One letter.
The pen has two coordinate signals: x(t) and y(t). The blue wave controls its left–right movement; the coral wave controls its up–down movement. Each signal is a sum of waves. Follow their guides to see the two coordinates trace the f from our wordmark.
The number of components increases automatically. Move the slider to take control.
Each circle carries a rotating arm; its endpoint carries the next circle. Blue builds x, coral builds y. Their waves feed the guides that meet at the yellow pen. Tiny circles are hidden, but every selected term contributes.
How the Fourier reconstruction works
The outline of the standalone f in the fifteen02 wordmark is sampled at 256 equally spaced positions along its closed boundary. Treating each position as x + iy lets a discrete Fourier transform find the rotating components. Each component has a radius (amplitude), starting angle (phase), and rotation rate and direction (frequency).
We keep the largest components first. Adding components reduces root-mean-square error over the original sample positions. It need not improve every individual point. With all 255 nonconstant components, the reconstruction matches the centred samples up to floating-point precision; between those samples it remains a trigonometric interpolation. We split the selected complex series into real x and y series using conjugate frequency pairs. Each coordinate has its own circle chain: the blue chain ends on its horizontal axis and the coral chain is turned to end on its vertical axis. The component slider counts terms in the original complex series; splitting them can produce a pair of circles in each coordinate chain. Each wave’s current value determines the matching coordinate of the pen. Time runs towards each wave’s moving endpoint; their differently oriented time axes allow the guides to meet at the drawing.
The outline is traced from the existing logo asset’s alpha boundary. This follows the edge of the printed letter, rather than a handwriting stroke. The letter is centred, so its constant position term is zero. Animation slows the reconstruction down for explanation; it does not show computation time. Discrete Fourier transform reference.