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.
What is the geocentric (Ptolemaic) model?
The geocentric model places Earth at the centre of the universe with the Sun, Moon and planets all circling it; Ptolemy worked it into a geometric system in the Almagest that could actually predict planetary positions. It does yield a planet's direction in the sky — the problem is not that it fails outright, but that planetary paths sprout loops that need epicycles, eccentrics and equants to fix.
What are epicycles and deferents?
The deferent is a large circle centred on Earth; the epicycle is a smaller circle whose centre rides along the deferent, and the planet moves around the epicycle. Move the origin of the same set of planetary positions from the Sun to Earth — subtracting Earth's position from every coordinate — and the outer planets' paths grow a chain of small loops by themselves. Those loops are the epicycle, and they come from Earth's own orbit.
Why did the heliocentric model replace the geocentric one?
Because with the Sun at the centre the planetary paths become simple: retrograde motion needs no extra circle, and the outer planets' one-year cycle and the way Mercury and Venus stay near the Sun are all explained by Earth's orbit. For naked-eye positional accuracy, though, the early heliocentric model was not immediately better than a well-tuned Ptolemaic one; it was Galileo's telescope — the full set of phases of Venus, Jupiter's own moons — and, in 1838, the first stellar parallax that supplied direct evidence of Earth's motion around the Sun.
Draw planetary motion from Earth and loops appear in the path
The same planetary positions form simple orbits around the Sun and loops around Earth. The difference begins with where the coordinate origin is placed.
First decide where the map's origin sits
With the Sun as origin, planets move along heliocentric orbits. Move the origin to Earth and Earth's position must be subtracted from every planet, changing the shape drawn on the page.
Nothing physical switches when the origin slider moves. Planet-to-planet distances and the directions actually seen from Earth remain the same; only the coordinate system used to describe them changes.
The small loop grows from Earth's own orbit
An outer planet's geocentric path contains repeated loops. One part comes from that planet's solar orbit and another from subtracting Earth's motion; the second contribution produces the epicycle-like circle.
Geocentric coordinates can still predict sky directions. The heliocentric description is valuable because the paths become simpler and relationships between orbital distance and period share one physical explanation.
The heliocentric model draws far fewer circles and gives the periods a common cause
Geocentric coordinates can still produce a planet's direction in the sky; the problem is not that they are unusable. With the Sun at the center, planetary paths become simple, and patterns such as longer periods for greater distance from the Sun fit into one physical explanation.
Slide the origin slowly back to the Sun and the loops on the geocentric chart collapse into plainer orbits. That is where the heliocentric description earns its keep: the observations have not changed, but far fewer shapes need a separate explanation.
The same geometry, two origins
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.
That loop on the geocentric plot is the few weeks a planet spends moving backwards across the night sky. The retrograde lesson projects the same positions onto the background stars, and lists each planet's next retrograde dates.
Where the epicycle comes from
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.
This is not what Ptolemy actually built
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.
So why did it lose?
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 Venus-phase evidence comes with real numbers in the greatest-elongation lesson: at every greatest elongation of the past sixteen years, Venus was between 49% and 51% lit.
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.
Where the epicycle comes from
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
FAQ
Why does an epicycle appear when the origin moves to Earth?
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.
Was the Ptolemaic model just one Earth-centred circle?
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.
Was the heliocentric model more accurate from the outset?
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.
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.