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From: bs@faron.UUCP (Robert D. Silverman)
Newsgroups: net.puzzle,net.math
Subject: Re: Polar Bear Problem Sequel
Message-ID: <374@faron.UUCP>
Date: Tue, 29-Oct-85 12:45:06 EST
Article-I.D.: faron.374
Posted: Tue Oct 29 12:45:06 1985
Date-Received: Thu, 31-Oct-85 23:32:34 EST
References: <855@whuxlm.UUCP> <593@hou2c.UUCP> <373@faron.UUCP>
Organization: The MITRE Coporation, Bedford, MA
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Xref: linus net.puzzle:1019 net.math:2079

> > it's almost true everywhere - almost.
> 
> Do you really mean 'it's almost true everywhere' or do you mean 
> 'it's true almost everywhere'  ?
> 
> I hate to clue everyone in but:
> 
> IF your answer means that the Borel measure of the set of starting points
> is 1 you're wrong. It is zero. The set of starting points is that set
> such that the radius of a great-circle running E-W is the same as that
> of another great-circle running E-W which is 1 mile south. The only place
> this happens is the great-circle 1/2 mile north of the equator. Moving
> 1 mile south places you on the great-circle 1/2 mile south of the equator
> and this obviously has the same radius as the original circle.
> Thus, of the entire set of great circles (cardinality C) only 1 satisfies
> the conditions (i.e. measure is zero)
> 
> Postings which claim the circles 1 mile north of the equator are solutions
> are wrong. This is easy to see because lines of longitude are closer
> together 1 mile north of the equator than they are at the equator. Thus,
> if you travel 1 mile south to the equator, 1 mile west, and then 1 mile
> north you will be closer than 1 mile to your starting point.
>  
> At latitude 90-theta, an East-West great circle has radius 2 PI r sin(theta)
> where r is the Earth's radius. Why can't people do simple high school geometry?
>  
> Thus, it's FALSE almost everywhere.
> 
> Bob Silverman   (they call me Mr. 9)

There are also some 'peculiar shaped' solutions near the south pole. One walks
a mile south. Then walking a mile west wraps one some multiple N times around
a great circle. This does not bring one back to the point where one started
the East-West walk but rather leaves one at some other point on that circle.
Then walking 1 mile north places one back on the original great circle, only
one mile east.  Note that N is not an integer.

				-------  <-- 1st great circle distance = 1 mile
				|     |
	       1 mile north --> |     |	<-- 1 mile south
				|     |
				/     \	<-- imagine folded 90 degress toward you
			       /       \
			       |        |
			       \        / <-- 2nd great circle
				--------
 
The exact set of starting points would be a tedious, but fairly simple problem
in trigonometry to work out.
 
An equation for the simple case (where one does not wrap around each circle
more than one full turn) is:

		1			2 PI r sin( 90 - t1)  - 1
	------------------     =     -------------------------------
	2 PI r sin(90 - t2)             2 PI r sin(90 -t1)

One simply finds angles t1 and t2 that satisfy this equation.


Bob Silverman  (they call mr Mr. 9)