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The following figure shows how epicycles work.
Instead of using just one sine and cosine to calculate a position on the locus of the epicyclic curve (and getting an approximation to a circle, we add a series of sines and cosines. Each additional term in the sum multiplies our basic iteration angle and the we assign a different radius (a "harmonic amplitude") to each term, getting two summed series which are used to calculate the next X and Y coordinates of the curve:
X = ∑ Rn cos(2π nθ)
Y = ∑ Rn sin(2π nθ)