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May 20th, 2014, 10:55 AM   #1
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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)
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May 20th, 2014, 04:45 PM   #2
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The shortest distance between two points is a straight line.
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May 20th, 2014, 05:07 PM   #3
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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 finger-in-the-air though.

Last edited by v8archie; May 20th, 2014 at 05:09 PM.
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May 20th, 2014, 05:14 PM   #4
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Quote:
Originally Posted by Hiks7888 View Post
does anyone know something similar or saw some works about it?
Google: lost in the forest problem
Here's a paper with a solution.

It appears that your suggestion is not optimal, but it is quite close.
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May 20th, 2014, 09:28 PM   #5
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The most rational way would be to travel in a circle where radius of circle is 1.
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