that is one place where Parelli differs quite markedly from most of the other NH trainers/clinicians. He has student teach a horse to go in circles around the human, while the human stands still. The idea being that the hrose should take the gait the handler tells him to, and keep at it until told to change.
My trainer, who would be closer to the Branaman/Dorrance camp , feels that this is rather pointless. She has us lunge in a way that our body is always riding the horse. So, if the horse is walking, our body reflects the energy of a walk, and we circle in a very small circle, always facing the horse, always watching t see the horse's mental focus and his manner of walking, not just that he stays in that gait. if we want a trot, we get his attention (like a halfhalt) an with the increase of energy in our own bodies, we ask the hrose to match us and thus trot off. To bring him back to walk, again, get his attention, decrease the energy in our body and expect him to slow to match it. If we stop "riding", it means we stop walking and stand. That means the horse should also stop , to match the stillness in our bodies.
So, having the horse circle around where our body is the same stillness that it would be if we wanted them to stand still,makes no sense, to me.