  In the framework of the planar and circular restricted three-body problem, we
consider an asteroid that orbits the Sun in quasi-satellite motion with a
planet. A quasi-satellite trajectory is a heliocentric orbit in co-orbital
resonance with the planet, characterized by a non zero eccentricity and a
resonant angle that librates around zero. Likewise, in the rotating frame with
the planet it describes the same trajectory as the one of a retrograde
satellite even though the planet acts as a perturbator. In the last few years,
the discoveries of asteroids in this type of motion made the term
"quasi-satellite" more and more present in the literature. However, some
authors rather use the term "retrograde satellite" when referring to this kind
of motion in the studies of the restricted problem in the rotating frame. In
this paper we intend to clarify the terminology to use, in order to bridge the
gap between the perturbative co-orbital point of view and the more general
approach in the rotating frame. Through a numerical exploration of the
co-orbital phase space, we describe the quasi-satellite domain and highlight
that it is not reachable by low eccentricities by averaging process. We will
show that the quasi-satellite domain is effectively included in the domain of
the retrograde satellites and neatly defined in terms of frequencies.
Eventually, we highlight a remarkable high eccentric quasi-satellite orbit
corresponding to a frozen ellipse in the heliocentric frame. We extend this
result to the eccentric case (planet on an eccentric motion) and show that two
families of frozen ellipses originate from this remarkable orbit.
