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May 20th, 2014, 10:55 AM  #1 
Newbie Joined: May 2014 From: Russia Posts: 1 Thanks: 0  Curves that couldn't be inscribed in rectangle
Problem: tourist get lost in the forest. Forest is rectangle with width = 1 and height >>> 1. So what curve will be the shortest universal way out? So in this problem we need to find shortest curve which couldn't be inscribed into rectangle. It seems to me that the shortest way is two of three curves of Reuleaux triangle, drawn around equilateral triangle with heigth = 1. Now my questions: first of all, am i correct? and second, does anyone know something similar or saw some works about it? I'll be really pleased if someone helps me) 
May 20th, 2014, 04:45 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,963 Thanks: 1148 Math Focus: Elementary mathematics and beyond 
The shortest distance between two points is a straight line.

May 20th, 2014, 05:07 PM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,683 Thanks: 2664 Math Focus: Mainly analysis and algebra 
But the point of the problem is that the person doesn't know in which direction he is travelling, nor where he is in the forest. If he heads off in a straight line, he may be travelling parallel to the longest side (or worse). A curved path will avoid this problem. I think that the answer might reasonably be a curve that can be inscribed in the rectangle. Or rather, one which will touch both sides regardless of orientation. So the shape might be right, but I'd probably want the side of the equilateral triangle to be equal to 1. This solution seems right for the shortest curve that is guaranteed to get him out, but I wonder if there is a better solution for the shortest expected distance to travel. All this is fingerintheair though. Last edited by v8archie; May 20th, 2014 at 05:09 PM. 
May 20th, 2014, 09:28 PM  #5 
Senior Member Joined: Nov 2013 From: Baku Posts: 502 Thanks: 56 Math Focus: Geometry 
The most rational way would be to travel in a circle where radius of circle is 1.


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curves, inscribed, rectangle 
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