Ptolemy's Epicycles

Any closed curve can be approximated by a sum of rotating vectors (Fourier series). Ptolemy's epicycles were an astronomical model; mathematically they are exactly a DFT. Each "planet" orbits on a circle that itself orbits another circle. Add more epicycles to approach any target shape.

Number of epicycles: 5

Speed: 1.0x

Target shape:

Epicycles: 5

Freq range: —

Phase: —

The DFT coefficients cₙ = ∫f(t)e^{-2πint}dt give amplitude and phase for each epicycle frequency n. Ptolemy used these implicitly for planetary motion.

Adding more epicycles = more Fourier terms = better approximation.