  Even professional baseball players occasionally find it difficult to
gracefully approach seemingly routine pop-ups. This paper describes a set of
towering pop-ups with trajectories that exhibit cusps and loops near the apex.
For a normal fly ball, the horizontal velocity is continuously decreasing due
to drag caused by air resistance. But for pop-ups, the Magnus force (the force
due to the ball spinning in a moving airflow) is larger than the drag force. In
these cases the horizontal velocity decreases in the beginning, like a normal
fly ball, but after the apex, the Magnus force accelerates the horizontal
motion. We refer to this class of pop-ups as paradoxical because they appear to
misinform the typically robust optical control strategies used by fielders and
lead to systematic vacillation in running paths, especially when a trajectory
terminates near the fielder. In short, some of the dancing around when
infielders pursue pop-ups can be well explained as a combination of bizarre
trajectories and misguidance by the normally reliable optical control strategy,
rather than apparent fielder error. Former major league infielders confirm that
our model agrees with their experiences.
