Fourier Analysis: Drawing Llamas with Circles

The Fourier transform converts signals from the time domain to the frequency domain, and it can also draw pictures. By chaining rotating vectors tip to tail, each spinning at a different frequency, you can trace any closed curve—from a simple line to a fish to a llama using 1024 circles. This article explains the math behind the Fourier series and transform, covering complex numbers, linear algebra, and inner products, to show how these epicycles recreate your sketches.