The course lying before the assembled runners heads up a steep hill and
bends sharply to the left. When the gun goes off, many of the more seasoned
runners head to the left, towards the inside of the curve. They are
"running the tangents". Unfortunately for them, the course beyond the crest
of the hill bends sharply back towards the right. They would have done better
to have run straight down the middle of the road.
All runners have been told to "run the tangents" at some point in their
career. How can this advice be justified? How can it be followed?
With some minor assumptions, the problem boils down to one of minimizing
distance. Suppose the "course" is represented as the interior of some
region in the plane, with the start and finish being marked points within
that region. The problem is to find, among all curves that connect the
start to the finish and remain everywhere inside the region, the one
of minimum total length.
It turns out that such a minimal curve consists only of straight sections
and sections that hug the boundary. One should never run a curved path while
in the interior part of the course. This certainly supports the received
wisdom of runners and seems quite intuitive upon reflection, but the
mathematics needed to derive this result are suprisingly sophisticated.
(The methods of the calculus of variations can be used.)
If the finish line is not in the line of sight, the runner should be
heading directly towards a point on the boundary of the course. But
which point? Unfortunately, this question has no easy answer.
To see some of the difficulties, consider a very simplified model of a
race course. The course consists of consecutive sections of one of three
types: straight sections(S), curves to the right(R), and curves to the
left(L). A great variety of courses can be assembled from these pieces,
just as the similar pieces used in model railroads can be used to build a
great variety of layouts. Thus, one possible course might be described as
RLSSLLRS.
Things get interesting when one considers how to combine the effects of
consecutive sections of the course. For example, an R followed by an S
or by another R could be treated as just one big R. On the other hand,
an L following an R would tend to cancel it, so that the two together could be
considered an S. The following 9 rules for combining consecutive sections
seem reasonable: RL=S, RS=R, RR=R, SR=R, SS=S, SL=L, LR=S, LS=L, LL=L.
Thus, to get "the big picture", a runner might apply these rules
successively to reduce the entire course to a single left, right, or
straight tendency, and set off on the first leg accordingly. For example,
the course LRLS could be reduced from left to right as follows:
LRLS --> SLS --> LS --> L
So a racer on this course would deduce that, overall, the course is
bending to the left, and he should head for the inside of the first
left curve.
But wait a minute! What is a runner to do with a course like LRRRL? One
person, applying the above rules from the inside out (LRRRL --> LRRL -->
LRL --> SL --> L) would conclude that the course tends towards the left,
while another working from the outside in (LRRRL --> SRS --> RS --> R) would
reach the opposite conclusion!
It turns out that the proper way to analyze a course is from the finish back
to the start. (This is a basic mathematical principle in optimization known as
"backward induction.") So, in the above example, the combination rules should
be applied as follows L(R(R(RL))). Thus, the start of the course looks like
an LR=S, and the runner at the beginning should head straight down the road.
From there, the course looks like RRRL, which reduces right-left to
R, so the runner should head towards the inside of the right curve, etc. Thus
the proper (optimal) way to run the segments of this course is SRRS.
Obviously this is all a great oversimplification, but it reveals that a full
analysis of the original problem can be no less intricate than this.
As long-time readers of this group know, I enjoy inventing offbeat running
events. Here is my latest, the running equivalent of "compulsory figures" in
figure skating: It is held in a large parking lot, on which is outlined in
chalk an intricate, serpentine, running course. It is a sprinters event: the
course is not long, and the object is run through it as fast as possible
WITHOUT STEPPING ON A BOUNDARY LINE. The latter point is crucial. One misstep
and the vigilant officials will flag the runner as disqualified. In such
an event, like the 100m dash, every tenth of a second is important, and
the winner will be the runner who combines speed, agility, memory, and the
ability to find the shortest path. To make the latter aspect more interesting,
the runners will be given only 10 seconds to study a map of the course
beforehand.
Think of it as "fast-twitch orienteering."