Epicycles are multiple circular motions with different speeds and radii added together. Some combinations produce nice visual patterns.

Use the Quick Start parameters to generate some combinations for you. If you have no idea where to start, try N = 4, C = 1, G = 3, 5 or, 7. Varying D will then generate combinations of various radii.

Here are the details. Epicycles can be considered as parametrized curves in the complex plane of the form
z(θ) = NΣi = 0R_i exp(i S_iθ).

When all S_i are real and rational multiples of each other, the curves are periodic and closed. For our purposes, 0 ≤ R_i ≤ 1, S_i ∈ ℕ₀, 0 ≤ θ ≤ 2π.

As you add cycles and set R_i and S_i—through either Quick Start or Manual, you will see the resultant curves in the box.

Note that changing any value in the Quick Start box overrides all the current Manual settings.

About Quick Start: The parameter N in the Quick Start box is of course the number of cycles. The meaning of the remaining parameters in the Quick Start box is left out; they are there to just generate R_i and S_i combinations. You will figure it out though, if you play with them for a while.

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