I've read it several places that the apparent retrograde motion of planets (during which they seem, as viewed from Earth, to move in the opposite sense of their normal "direct" orbital motion against background stars at infinite distance) occurs between the two quadrature points (at which the planet-Earth-Sun angle is 90°). I have always assumed that "between" is an approximation, since, at quadrature, though the Earth's motion is contributing nothing to the planet's apparent motion (since it is moving directly along the line of sight to the planet), the planet's true motion is providing apparent "direct motion", so that retrograde motion, though bounded by the quadrature points, does not begin or end there.
I've never been able to prove this to my satisfaction, and often come across descriptions that make me wonder whether I've had it wrong all along.
Some of these sources would seem to be quite authoritative, such as a translator's footnote to Copernicus, in which it is stated that the angular extent of a superior planet's retrograde motion observed from Earth is defined by the tangents to the Earth's orbit that pass through the planet:
While these tangents certainly bound the angular extent of retrograde motion — in fact, they define the retrograde (and direct) motion of an unmoving planet (since they correspond to the maximum parallax for the planet seen from Earth) — isn't the actual extent of retrograde motion smaller, for the reasons stated above?
How can it demonstrated geometrically (assuming circular orbits with common centers and uniform angular velocities, and given periods and radii for those orbits for Earth and the Planet) what the rate of change of the apparent angular position of an orbiting planet is as a function of the planet-Earth-Sun angle?