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Michael K. McBeath, Thomas G. Sugar, Zheng Wang; Baseball fielders utilize a rule of constant cotangent change to navigate to catch ground balls. Journal of Vision 2004;4(8):157. doi: https://doi.org/10.1167/4.8.157.
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When baseball fielders catch fly balls they run along a path that keeps the image of the ball rising both at a constant speed and along a straight line, upon a vertically-oriented virtual projection plane. Mathematically this is equivalent to maintaining a constant rate of change in the tangent of the vertical optical angle, a (i.e. ?tana/?t = Constant), and a matched rate of change between the respective vertical and lateral optical angles, α and β (i.e. ?α/?β = Constant). In tests of these models, the typical fit accounts for over 95% of the variance in optical ball movement, which supports use of a vertically-oriented projection plane. In the present study, we examined behavior of fielders catching ground balls in a 3-D motion capture room that provided cm-resolution accuracy of head, body, and ball position over a 25 by 15 foot area. The Vicon eight-camera set up creates point-light figures of the moving fielder and ball at a 120 Hz frame rate. We tested three fielders differing in skill level, who pursued rolled balls that varied in speed and angle. The findings support that, as with fly balls, fielders keep the image of the ball moving at a constant speed along straight line, but here the virtual projection plane was horizontal. Mathematically this is equivalent to maintaining a constant rate of change in the cotangent of the vertical optical angle, a (i.e. ?cota/?t = Constant). With these precisely controlled measurements, the model fit typically accounted for over 99% of the variance in optical ball movement, which strongly supports use of a horizontally-oriented projection plane. Measurements using fielders with head-cams in a larger outdoor setting further confirms these findings. The pattern of results is consistent with a unified fielder theory in which rotated versions of the same optical control heuristics of speed constancy and linearity are used to navigate to intercept fly balls headed above the horizon and ground balls headed below it.
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