Article: 125079 of sci.geo.satellite-nav From: davem@cs.ubc.ca (Dave Martindale) Newsgroups: sci.geo.satellite-nav Subject: Re: How far North will GPS work? Date: 4 Dec 2000 17:26:42 -0800 Organization: Computer Science, University of B.C., Vancouver, B.C., Canada Lines: 51 Message-ID: <90hg8i$14h$1@lily.cs.ubc.ca> References: <3A2C1096.45A73382@netscape.net> <90h69d$dre$1@lure.pipex.net> <3A2C2BAA.BED4E848@dircon.co.uk> NNTP-Posting-Host: lily.cs.ubc.ca X-Trace: mughi.cs.ubc.ca 975979603 22150 142.103.9.66 (5 Dec 2000 01:26:43 GMT) X-Complaints-To: usenet@cs.ubc.ca NNTP-Posting-Date: 5 Dec 2000 01:26:43 GMT Path: news.meer.net!nntp1.ba.best.com!news2.best.com!news-hog.berkeley.edu!ucberkeley!nntp.cs.ubc.ca!cs.ubc.ca!not-for-mail Xref: news.meer.net sci.geo.satellite-nav:125079 johnb@dircon.co.uk writes: >An explorer is at point X on the surface of the earth. >He walks 1 km South, then 1 km East, and then 1 km North, and he is back >where he started at point X. >Questions: >1. Where is point X? >2. How many correct answers are there? >3. Would a GPS give his position accurately at point X? Three classes of answers for #1: 1 The explorer starts at the north pole. All directions are south from there. After walking 1 km in any direction, he is 1 km south of the pole. Walking any distance east takes him to some other point which is on the same line of latitude, and thus *still* 1 km south of the pole. Walking 1 km north returns him to the pole. 2 The explorer starts at a point 1 + 1/(2*pi) km from the south pole. When he walks 1 km south, he is now 1/(2*pi) km from the south pole. Then he walks 1 km east along a line of latitude. Now, the diameter of the complete circle at this latitude is exactly 1 km (because the radius of the circle is 1/(2*pi), so walking 1 km east brings him right back to the point where he started travelling east. Then 1 km north returns him to his starting point. (Note that the true answer to this isn't quite 1/(2*pi), since that's a flat-earth approximation. But this is all so close to the pole that it will be very accurate). 3 There are a whole class of solutions very much like #2, but which involve southern-most circles which are integer sub-multiples of 1 km, and where walking 1 km east takes the explorer some integer number of times around the circle, returning to the same spot. For example, if the starting point is 1 + 1/(5*2*pi), after walking 1 km south the explorer is 1/(5*2*pi) from the south pole, and the diameter of the circle of that radius is 1/5 km. So 1 km east takes the explorer 5 times around the circle to the same spot... So, to answer #2: Solution #1 provides just one point. Solution #2 provides a single circle containing an infinite number of points. Solution #3 provides an infinite number of circles (but they are countable because they are numbered with integers), and each circle has an infinite number of points on it which satisfy the puzzle. As for question #3, which of infinitely many point X's did you mean? And what does "accurately" mean to you anyway? Dave