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February 24th, 2007, 12:32 AM   #1
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-------shortest distance-----------

..can any1 help
A geodesic is (in mathematics) the shortest line between two points on a
mathematically defined surface (as a straight line on a plane or an arc of a great circle on
a sphere).
You are given a 1x1x2 cuboid. You are required to determine 2 points on the surface of
the cuboid such that they have the maximum possible geodesic.


Hint: The required distance is not the geodesic between the mid points of the two faces as commonly misunderstood.
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February 24th, 2007, 03:50 AM   #2
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I have no proof of this, but I would imagine that the two points are opposite corners. The distance between them is √10≈3.1623 as opposed to 3, the distance between the centers of 1x1 faces.
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February 24th, 2007, 10:34 AM   #3
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Quote:
Originally Posted by roadnottaken
I have no proof of this, but I would imagine that the two points are opposite corners. The distance between them is √10≈3.1623 as opposed to 3, the distance between the centers of 1x1 faces.
actually i don see how u get sqrt 10 .....please explain
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February 24th, 2007, 01:52 PM   #4
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Cut the surface and "unfold" it to a flat surface. There are various ways of doing that. Now join the opposite corners with a straight line. Check whether any different way of unfolding results in a shorter line. Obviously, reject any such line that leaves the surface. Calculate the length of the shortest line using Pythagoras's theorem.
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February 24th, 2007, 06:09 PM   #5
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Obviously, calculating the length of the geodesic between two opposite corners does not show that that is indeed the longest geodesic. The next step would be to rigorously prove that it is.
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February 25th, 2007, 07:54 AM   #6
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There are a limited number of ways to flatten out the surface that need to be considered. Even so, a rigorous proof seems a bit tedious.
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March 4th, 2007, 10:01 AM   #7
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would it be less boring of a proof, if one shows that if any point has length longer than the diagonal corners, one would lie outside of the cube?
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