Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.2 9/18/84; site faron.UUCP Path: utzoo!linus!faron!bs 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 Lines: 61 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)