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Topic: Inscribed Triangle (Read 1924 times) |
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SWF
Uberpuzzler
Posts: 879
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Re: Inscribed Triangle
« Reply #25 on: Sep 9th, 2007, 11:00pm » |
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Imagine a tiny loop of rubber band much smaller than the triangle that is forced to have frictionless contact at a point on each of the three sides of the triangle, and the contact points allow slip of the band so that tension is the same throughout the rubber band. The rubber band must be stretched to make it touch each side, but will take on the shape of the inscribed triangle of minimum perimeter to minimize the strain energy due to stretching.
At any of the points where the rubber band touches a side of the triangle, for it to be in equilibrium, the angle "incidence" and "reflection" must be the same. Thus, the shape of the minimizing inscribed triangle is the reflecting ray of light path as described by Eigenray.
The same should be true not matter what the shape within which you are trying to inscribe a polygon of minimium perimeter- although there are likely to be several shapes having a local local minimum (or maximum) perimeter. You are likely to get some of the contact points to coalesce. For example, trying to find a minimum permeter quadrilateral might degnerate into a triangle, and there should alway be at least one such shape (otherwise you would have perpetual motion).
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