Some scholars say that Copernicus was influenced in his astronomical theory by 13th century Persian mathematician Nasir al-din al-Tusi, who discovered (or re-discovered) a principle for converting angular motion to reciprocating linear motion. Watch the neat video below to see the basis of this motion conversion
The ‘Tusi couple’ is covered by this wiki page, and has this simpler animation to show the principle. But in relation to Copernicus, an his supposed ‘borrowing’ of this device to explain the apparent equant motion of planets as viewed from Earth there is also reference to a paper by I. N. Veselovsky, “Copernicus and Nasir al-Din al-Tusi”, Journal for the History of Astronomy, 4 (1973): 128-30
In that paper, Veselovsky notes that Proclus, the 5th century Neoplatonist philosopher, wrote a trestise on Euclid’s geometry which also contains the circular to linear motion principle, and that a passage from Proclus is repeated in Copernicus’ work. It’s possible Proclus included this aspect of Euclid because it was highlighted by a slightly earlier Greek scholar, Eudoxus.
Those interested in the shift from the classical and Arabic geocentric world to the heliocentric model of Copernican astronomy can find a useful passage in this google books link to Toby Huff’s work ‘Intellectual curiosity and the scientific revolution: A global perspective’.
With reference to the post on resonance rings in galaxies immediately preceding this one, consider the path taken by the point traveliing along the line in the wiki animation if the outer circle and the line were also revolving, at half the rate of the inner circle. It’s a bit hard to visualise, but I think it might make a figure 8 as it completes an ‘orbit’. Anyone got the time to model it?