Building a Bridge
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Explanation

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Copyright © 1996-2017 Alexander Bogomolny

 

This one is a basic optimization problem. It's well known and serves as an easy illustration of the usefulness of the simplest of geometric transforms - translation.

Two points A and B are given on opposite sides of a strip defined by two parallel lines. The strip represents a river, and the points two cities on the opposite sides of the latter. The problem is to build the shortest possible road between A and B, assuming that the land parts of the road are straight line segments and the bridge is perpendicular to the banks.

Let C be a point on the upper bank and C' its mate on the lower bank, so that CC' is perpendicular to both lines. CC' defines a vector V and a translation transform in the plane. It is clear that the length of V enters all possible choices of C on the upper bank. The problem is thus equivalent to minimizing the "land" sum AC + C'B.

Translate point B by -V to obtain B'. By the triangle inequality,

AC + CB' ≥ AB',

while CB' = C'B. Therefore, the shortest route is defined by the position of C where the line AB' crosses the upper bank.

Clearly, we could have translated A by V to A' and considered intersection C' of A'B with the lower bank. The result would have been the same.

Note: the problem admits extensions to more than one river seperating the two cities. The case of two rivers is discussed elsewhere at the site.

References

  1. I. M. Yaglom, Geometric Transformations I, MAA, 1962

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