Every planet's place in the sky is the sum of two rotating circles — the deferent and the epicycle. Watch the retrograde loop appear on its own.
This is not the FFT parlor trick that traces any shape from enough circles. It is the real thing:
a planet's apparent position equals the heliocentric planet vector minus the heliocentric Earth vector —
exactly two rotating circles summed. That is the original Ptolemaic deferent + epicycle, which is a two-term Fourier series.
Retrograde is not the planet reversing — it is the pen looping as the two circles beat against each other.
Positions computed with Swiss Ephemeris; the geometry is fact. Scalar Flower · A Research Project.