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Hi all-
I've skipped some messages, so mebbe the problem changed.
But the last version I saw had an east-directed component (line of
constant latitude). Latitude lines are not -with one exception -
great circles (geodesics).
John Denker wrote, in part:
Here's the outline:
1) The legs of the triangle are by definition geodesically
straight.
2) That means each leg is a piece of a great circle, since
we are dealing with a spherical world.
Have I missed something?
Wait a minute.
Didn't we begin by walking East a mile or so from a pole?
That's not a Great circle.
When did I miss a turn off track?
Dr. Livingstone starts out at a place which we call
Point A. He then undertakes a journey consisting
of three legs:
-- he travels precisely southward for one mile;
-- then he turns and travels precisely eastward
for one mile;
-- then he turns and travels precisely northward
for one mile.
He discovers that as a result of this journey, he
has returned to Point A.
Note: He travels by airship, at an appropriate constant
altitude, so you don't need to worry about obstructions
or other nonidealities.
The questions are:
1) Where is Point A?
2) Are you sure? How do you know?
If you are old enough, the question is more or less identical to a
widely spread set of riddles that were spread around in the
seventies. And the description of the problem was followed by the
question: What color are the bears?
Suppose we straightened out the 2nd leg of the path so *all three*
legs of the path are geodesically straight and the length of the 2nd
leg is the same as the length of the 1st and 3rd legs. The whole
closed path is now an equilateral triangle as inscribed onto the
spherical surface. The problem is to find a formula for the measure
of the interior angle of such an equilateral triangle as a function
of the length s of the sides of the triangle (conveniently in units
of the sphere's radius). A few hints are that 1) the value of the
formula must boil down to 60 deg in the limit of s becoming a
zeroth fraction of the sphere's radius, 2) the value of the formula
becomes 90 deg when s is 1/4 of the circumference of the sphere,
3) the maximum size triangle occurs for a great circle with 3
equally-spaced vertices (120 deg apart from each other) on it with
the interior angle at each vertex being 180 deg across the vertex
and each side having a length s of 1/3 of the sphere's
circumference, and 4) the messy intermediate math eventually
simplifies to a relatively simplified formula in the general case.
For a lot of extra credit points you can also find the proper
formula for the *area* of this spherical equilateral triangle in
terms of the length s of the sides of the triangle (making sure
that the formula boils down to all the correct values for the
variously known special cases).
Or how about this one:
From a fixed point, head due South, continue in a straight line
for one mile: turn due East and continue in that direction for one
mile, then turn onto a due South heading and continue straight for
one mile, to return to your start point.