An area in the shape of a square 10 units on a side needs to be mowed. The mower, which only goes forward, has a mowing "footprint" that is a unit square, and turns about the center of its footprint.
An optimal mowing plan is sought. A mowing plan designates a starting position and from there gives a complete mowing path. An optimal mowing plan is one closest to a straight line in the sense that the sum of all the changes in the mower's angular direction is minimized, each such change taken as the minimum possible positive value. The mower is not impeded by the border of the square and can travel without difficulty outside it as well as inside it.
Is "spiral" better than "back and forth," and what about a "diagonal" plan?
Depending on how much "extra" work you want to do, you could mow the lawn with as little rotation as you want, limited only by practicality and the curvature of the earth.
For example, starting at on extreme corner, mow parallel to the side, but continue well past the lawn boundary. Then rotate 1 degree so that the back end of the mower rotates towards the center of the lawn, and back up untilthe mower is one mower width perpendicularly towards the centre of the lawn. Un-rotate the 1 degree, and mow the second strip. Repeat until the lawn is completely mowed, and you've rotated a total of only 18 degrees. Of course, you could do better, if you wanted to mow more extra area outside the square.
At some point, the rotation caused by the curvature of the earth will begin to work against you, rotating you more about the center of the earth than parallel to the surface, but that is another problem altogether.
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