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fielders, fly balls, and formal loop specs

There is a brief communication in this
week's _Nature_* which claims fielders
repeatedly satisfy a simple constraint
problem when intercepting fly balls.

Their observation, tested in simulation
and against traces of real fielders, is
that fielders keep track of the rate of
change of the two angles relating their
position to the ball's.  The horizontal
change in angle is held constant, while
the vertical change is decreased.

As psychophysicists, they put it in the
following terms: the vertical decrease
means that the fielder's possible spots
are located on shrinking circles around
the ball, while the horizontal constant
ensures that the fielder will run in a
straight line towards the interception
point.  If the fielder obeyed the first
constraint alone, he might arrive at an
impractical solution requiring either a
discontinuous motion or a motion faster
than he could run.

As computer scientists, we would put it
in changed terms: the vertical decrease
as loop variant, the horizontal constant
as loop invariant.  The variant ensures
that the fielder intercepts the ball in
(pace Zeno) a finite time; the invariant
ensures that the fielder doesn't arrive
at an impractical solution requiring an
invalid access: either well outside of,
or not following the natural structure
of, his datatypes.


:: :: ::

* McLeod, Reed, Dienes, "How fielders arrive in time to catch the ball",
_Nature_ v426 20 Nov 2003