Drag the slider and the origin moves from the Sun to the Earth. Nothing is recalculated: every planet stays exactly where it was, and the loops that appear are what subtracting Earth's own motion looks like.
A position is a vector, and a vector needs somewhere to start from. Call Earth's position E and a planet's P. The heliocentric picture plots P; the geocentric one plots P minus E. Subtracting the same quantity from everything cannot change any distance between two objects or any angle between them, so both pictures predict identical positions on the sky. They are not rival predictions. They are the same prediction written in two coordinate systems.
That is why the control above is a slider and not a switch. Halfway along is a perfectly valid frame too, with the origin somewhere between the Sun and the Earth, and it is no more wrong than either end. What makes the heliocentric frame better is not that it is true and the other false. It is that in it the orbits are simple closed curves, and in the other one they are not.
So the epicycle is not a fudge factor. It is Earth's orbit, or the planet's, depending on which side of us the planet is. Its size and its period follow from that, and both are checkable:
Every superior planet has an epicycle period of exactly one year and a radius ratio of exactly 1/a. Ptolemy had no explanation for that coincidence. Copernicus needed none, because in his picture it is not a coincidence at all: it is the Earth going round once.
The diagram above is an idealised deferent-and-epicycle model, exactly equivalent to the heliocentric one. The Almagest is considerably more sophisticated than this, and pretending otherwise would be unfair to it.
It also has the property that the motion is not uniform and circular about anything at all, which is exactly what Ptolemy's critics objected to, Copernicus among them. The complaint was not that the model gave wrong answers. It was that it broke the rule it claimed to be obeying.
The observations that settled it came later. Galileo's telescope showed Venus running through a full set of phases, including a gibbous one, which requires Venus to pass behind the Sun and is impossible if it rides an epicycle that keeps it between us and the Sun. Jupiter's moons showed that not everything goes round the Earth. Stellar parallax, the direct proof, took until 1838.
The part worth keeping is the one the slider makes. A change of frame is free, and choosing the frame in which the description comes out simplest is a decision about description, not about truth. The planets do not know where we put the origin.
| Planet | a (AU) | Epicycle / deferent | Epicycle period |
|---|---|---|---|
| Mercury (inferior) | 0.387 | 0.387 | 0.24 yr |
| Venus (inferior) | 0.723 | 0.723 | 0.62 yr |
| Mars | 1.524 | 0.656 | 1 yr |
| Jupiter | 5.203 | 0.192 | 1 yr |
| Saturn | 9.537 | 0.105 | 1 yr |
| Uranus | 19.189 | 0.052 | 1 yr |
| Neptune | 30.07 | 0.033 | 1 yr |
Split P minus E into two pieces and you have a deferent and an epicycle. For a planet outside Earth's orbit, let the deferent carry a point out to P and let the epicycle vector be minus E: the epicycle is Earth's own orbit, seen from the wrong end. For a planet inside Earth's orbit the roles swap. The deferent carries a point to minus E, which is the Sun's direction from here, and the epicycle vector is P.
Ptolemy's planets do not ride a circle centred on the Earth. The deferent's centre is offset from us, and the centre of the epicycle sweeps out equal angles not about that centre but about a third point, the equant, placed on the far side of it. That device let the model reproduce a planet moving faster along one part of its path than another, and for a small eccentricity it is a remarkably good approximation to Kepler's second law, thirteen centuries early.
Not on accuracy. For naked-eye positions a well-tuned Ptolemaic model and Copernicus's original are hard to tell apart, and Copernicus still used epicycles, because he still insisted on circles. The case against the geocentric picture is that it has to put in by hand what the other gets for free: that one-year period in the table above, repeated separately for every superior planet, and the fact that Mercury and Venus are tied to the Sun's direction.
Related tools: Why planets appear to move backwards · Solar System Orrery · Greatest elongation
Orbital elements from NASA/JPL, "Keplerian Elements for Approximate Positions of the Major Planets" (Standish & Williams, 1992), valid 1800–2050. Positions are geometric — no light-time or aberration — because this is a diagram of the ecliptic plane, not a simulation of a telescope view.