# Building a Bridge

What is this about?

A Mathematical Droodle

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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

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

- I. M. Yaglom,
*Geometric Transformations I*, MAA, 1962

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