# Ivars Peterson's MathTrek

October 14, 2002

## Catching Flies

Archerfish and baseball outfielders appear to use different strategies to snag a projectile.

Archerfish (Toxotes jaculatrix) are famous for their unusual way of hunting insect prey. Upon spying an insect on a twig or a piece of foliage hanging above the water surface, the fish shoots it down using a strong, accurately aimed jet of water. Once dislodged by a shot, the insect tumbles down, following a roughly parabolic path.

Archerfish typically swim around in shooting parties. Often, more than one fish spits at the same target. And, when a bedraggled insect hits the water surface, it's first come, first served. So, it's to a shooter's advantage to be able to predict a waterlogged insect's landing place and get to it as rapidly as possible.

Experiments now suggest that an archerfish needs just a quick glance to judge where the dislodged prey will later hit the water and promptly moves in that direction well before the insect splashes down. "In contrast to other known examples of three-dimensional target interception in man and animals, archerfish can head straight to the predicted point of catch without the need for any further visual feedback," biologist Stefan Schuster and his collaborators at the Albert-Ludwigs-Universität-Freiburg in Germany report in the November Journal of Experimental Biology.

To study target interception in archerfish, the researchers installed five fish in a 600-liter tank. They tantalized the fish by offering flies that, when dislodged or released, never actually reached the water. To determine whether the fish needed to follow a tumbling fly's entire trajectory, they attached the fly to a thin thread so that it could fall for only a short time before being jerked to a stop. In this case, the fish kept on going, even though the fly was left dangling in the air.

In another experiment, the target fly was pulled at a constant speed along a horizontal glass plate suspended above the fish. The fish still converged on the point where the fly would have landed if it had also been allowed to fall.

These observations suggest that archerfish somehow "calculate" where a plummeting fly will land, based on a few ballistic parameters that fix the point of impact. In effect, the fish appear to rely on visual measurements, made in about 100 milliseconds, of the insect's initial height and horizontal velocity to locate the eventual landing spot. Whether archerfish learn from previous efforts or calculate each trajectory from scratch isn't known yet, however.

"In predicting the point of catch of their dislodged prey, archerfish outperform the strategies used, for instance, by baseball outfielders catching a fly ball," Schuster contends.

Nonetheless, it isn't so easy to ascertain what strategy a ballplayer uses to catch a fly ball, and the chosen strategy may depend on a player's experience, skill, or circumstances.

Most models of how fielders catch a ball invoke visual trajectory-tracking cues. For example, one model posits that outfielders run along a path that simultaneously maintains horizontal alignment with the ball and a constant change in the tangent of the angle of elevation of gaze from catcher to ball. As the ball rises, the tangent of that angle increases, but at a rate that depends on the selected running path. The model predicts that a player selects a straight running path and runs with constant speed to catch the ball.

A variant of this model removes the requirement that fielders run at a constant velocity. In this case, a successful fielder runs at a speed that maintains a relatively constant angle of gaze toward an oncoming projectile between 0° (looking straight ahead) and 90° (looking straight up) throughout its flight. If the angle reaches either 0° or 90°, the ball will be missed. Such a strategy ensures that fielders arrive at the right place at the right time but does not tell them where or when that is. In sum, fielders focus on the relationship between themselves and the ball, rather than predicting where the ball will land. They tend to position themselves so that a moving ball's path follows an apparently straight line.

In 1995, Michael K. McBeath and his coworkers proposed an alternative, two-dimensional model in which an outfielder selects a running path that maintains a linear optical trajectory (LOT) for the ball relative to home plate and the background scenery. In effect, a fielder runs along a curving path, adjusting his or her speed and direction so that the apparent trajectory of the ball stays in a straight line.

In general, these models require the fielder to watch the ball continuously throughout its flight. In one famous incident, however, something closer to the archerfish strategy may have been in play. That was Willie Mays' catch of Vic Wertz's fly ball in the 1954 World Series, where Mays apparently did not track the ball but still ended up in the right place at the right time.

In responding to this example, McBeath noted, "Our model addresses the behavior of recreational level players and leaves open the possibility that trained professionals might learn alternate strategies to enhance their performance."

Second, he added, "Mays' extraordinary behavior and deviation from the LOT model during this play may be part of the reason that it is considered by many to be one of the greatest in the history of the sport."

References:

Bower, B. 1996. Running gaze catches on with fielders. Science News 149(June 15):372.

Cipra, B. 1995. Catching fly balls: A new model steps up to the plate. Science268(April 28):502.

Gillies, M.F.O., and N.A. Dodgson. 1999. Ball catching: An example of psychologically-based behavioural animation. Available at http://www.cl.cam.ac.uk/users/nad/pubs/EGUK99MG.pdf.

McBeath, M.K., D.M. Shaffer, and M.K. Kaiser. 1995. How baseball outfielders determine where to run to catch fly balls. Science 268(April 28):569-573. See http://www.public.asu.edu/~mmcbeath/mcbeath.research/CatchFly/CatchFly.html. Letters commenting on the proposed model appeared in Science: 268(June 23):1681-1685. Follow-up articles offering other viewpoints appeared in Science: 273(July 12):256-260.

McLeod, P., and Z. Dienes. 1996. Do fielders know where to go to catch the ball, or only how to get there? Journal of Experimental Psychology: Human Perception and Performance 22(June):531-543.

______. 1993. Running to catch the ball. Nature 362(March 4):23.

Phillips, K. 2002. Archer fish outstanding in the outfield. Journal of Experimental Biology 205(November):2103. Available at http://jeb.biologists.org/cgi/content/full/205/21/2103/i.

Rossel, S., J. Corlija, and S. Schuster. 2002. Predicting three-dimensional target motion: How archer fish determine where to catch their dislodged prey. Journal of Experimental Biology 205(November):3321-3326. Abstract available at http://jeb.biologists.org/cgi/content/abstract/205/21/3321.

Westney, C. 2002. Archer fish never look back. Nature Science Update(Sept. 30). Available at http://www.nature.com/nsu/020923/020923-15.html.

A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the MAA book ;Mathematical Treks: From Surreal Numbers to Magic Circles. See http://www.maa.org/pubs/books/mtr.html.