At 09:48 PM 4/24/01 -0400, [EMAIL PROTECTED] wrote:
>... how would the topological features be taken into account when
>measuring the distances given in this puzzle. What if the sign was in on a
>mountain somewhere?
This is an intriguing question that is separate from the main one. To see
what effect a mountain would have, suppose the mountain is h miles
high. If you could glide off the mountain and follow a shortest path to
any of the cities, the total distance would not exceed sqrt(x^2 + h^2)
where x is the map distance to the city. (Alert readers will note I'm
applying the Pythagorean Theorem to a curved path, but this approximation
can be rigorously justified using the calculus expression for path
length.) When x > 4000 (as all the distances are) and h < 6 (as all
mountains are), then to an excellent precision sqrt(x^2 + h^2) = x +
h^2/(2x) and h^2/(2x) < 6^2/(2*4000) = 36/8000 = 0 (to the nearest mile),
so the mountain has no effect on distance.
If instead you flew at a constant altitude straight off the mountain,
reached one of the cities, and jumped straight down to the city center, the
distance x would increase to x' = x*(h + R)/R where R, the earth's radius,
is about 4,000 miles, and your jump would be close to h (depending on the
city's altitude). Those two effects would increase the distances
appreciably, so let's use a better value of h. We already know that there
is an excellent solution to the original problem, that it is unique, and
that it's in the middle of the U.S.. You won't find any points with h
greater than 3 miles there. Thus the maximum increase in distance would be
from x = 9092 to x' = 9092*(3+4000)/4000 + 3 = about a 10 mile
increase. That's well within the precision of the problem statement (which
reports all distances to the nearest mile, does not state what spheroid is
used, and expresses all terminal points as city names, which I believe have
an inherent +-10 mile ambiguity).
A calculation similar to the Pythagorean analysis above demonstrates that
having to navigate above or around a higher mountain or narrow mountain
chain along the route will not appreciably change the distances, either.
You can perform an analog computation to demonstrate these results: take a
relief globe and tightly stretch a wire between two widely separated
points. Anchor its ends firmly. It will still be easy to slip something
small under the middle of the wire or to wiggle the wire many tens or
hundreds of (scale) miles side to side. In so doing, the wire traces
innumerable paths of almost identical distance between the anchor points.
These kinds of simple geometric considerations indicate that topographic
altitude is not relevant when dealing with large distances when we have
only 0.1% precision.
If we instead suppose distances are measured as the car drives, then these
computations don't apply. The answer depends on the measuring
vehicle. Driving a car from Kansas to Johannesburg (with a bit of a ferry
along the way, I presume) will add some miles for the uphill and downhill
portions. An ant crawling the same way might have to navigate 100,000
miles of up and down to cover the same ground. At least we can establish
that these topographic distances are all *greater* than the geodesic
distances. It is easy to demonstrate that subtracting more than about 20
miles (0.2 to 0.4%) from each distance involved in the question (to convert
them to equivalent geodetic distances) produces a contradictory set of
data. Therefore, for this particular problem the distances *have* to be
very close to the true geodetic distances for there to be any solution of
interest at all.
This mapping puzzle, by the way, is relevant to certain kinds of GIS data
that come to us in the form "an observation was made at point X located at
distance a(1) from known point Y(1), at distance a(2) from known point
Y(2), ..., and distance a(n) from known point Y(n)." The analysis applies
to field measurements and raw GPS measurements, among others.
--Bill Huber
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