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?

It seems to this casual observer that a back-and-forth (my old asthetic choice) and a right angle spiral scheme both involve 18 90 degree turns, for a total of 1620 degrees (4.5 full circles).

Since the diagonal is 14.14 units, it seems to me that a true spiral would involve 7.07 full circles

If this unit area is on a sphere (like the earth), then we can do it all with 0 left or right turns, by doing a 3 dimensional spiral that would go around an almost-great-circle 11 times.  Of course, if we count following the curvature of the earth as a change in angular momentum, then this is not optimal  even if we had a very fast,  very fuel-efficient lawn mower.

I suspect that I have just traveled well outside the border of the square.

Posted by Steve Herman on 2005-02-13 13:17:39