Fourier Analysis: Drawing Llamas with Circles

Fourier analysis is used as a constructive way to draw closed curves by summing rotating vectors (epicycles): connecting vectors tip-to-tail whose radii, rotation rates, and start angles trace out a path such as a line, square, fish, or even a detailed llama generated with 1024 circles. Each rotating vector is encoded as a complex number capturing amplitude (radius), frequency (rotation rate), and phase (start angle), and increasing the number of circles yields progressively better approximations of the original path. Visual demonstrations show that wildly different component motions can combine to produce simple shapes and that modifying frequency or radius changes the drawn shape in predictable ways.

The mathematical substance connects the Fourier transform/series, complex exponentials, and linear algebra: transforming a time-domain path into frequency components produces spikes at component frequencies (e.g., sin(2π·3t) gives ±3 Hz), and Euler’s identity links those complex exponentials to sines and cosines. Viewing functions as vectors in a basis, the dot product/inner product measures how much each basis (frequency) contributes, so Fourier coefficients are projections that set each epicycle’s scale and phase; the inverse transform recombines them to reconstruct the curve. Continuous inner products generalize the discrete dot product to produce those coefficients for continuous paths.