Building Bridges

Two points A and B are separated by two pairs of parallel lines. A strip formed by parallel lines represents a river, and the points A and B are thought of as two cities separated by two rivers. 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 bridges over the rivers are perpendicular to their banks.

The problem is an extension of a similar one where the two cities are separated by a single river.

The banks of the rivers may be dragged. The effects are different when the straight lines are dragged near there endpoints and the midpoint.

Please note that, as stated, the problem does not always make sense. For example, if one of the points is inside a "river", there is not point in building the bridge here. Also, in the applet, the two lines forming a "river" play non-symmetric roles that can't be swapped. When one of the banks is dragged over the other, then, again, the diagram in the applet makes little sense.

Explanation

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 separated by two pairs of parallel lines. A strip formed by parallel lines represents a river, and the points A and B are thought of as two cities separated by two rivers. 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 bridges over the rivers are perpendicular to their banks.

Let A' be the point obtained from A by the translation in the direction of the bridge that would be built over the river nearest to A. (This direction is independent of the specific location of the bridge.) Of the two possibilities, we chose the one from A towards the river. Define B' similarly with respect to the second river. Join A' and B'. The intersections of A'B' with the banks of the rivers farthest from A' and B' respectively show the points where bridges should be constructed. If the bridges are denoted CC' and C''C''', then for the shortest route, AC||C'C''||C'''B.

The problem admits an extension to any finite number of rivers. The foregoing solution (for two bridges) needs to be modified only a little to be adapted for a more general case.

References

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