Efficient mowing technique

[RPM]

Glennmac,

Actually, the equilateral triangle proves the case - that was a great point to raise.

Imagine for a moment the tracks left by the mower if you were to mow in ever decreasing triangles. For any decent size lot (let's say half an acre or more) this will look pretty darned close to a set of triangles fitting inside each other.

Now hold that image of the triangles - decreasing in size - fitting inside each other. Now draw a straight line from each of the three corners of the original triangle to its middle. What do you have? Three smaller triangles that are mowed in parallel stripes. /w3tcompact/icons/cool.gif I've attached a diagram (5 mins in PowerPoint, so not so good) to try and explain.

 

[glennmac]

Wait a minute. This is getting dangerously close to a solution, but what have we proved? I count 14 turns in each pattern. Does this mean they are the same?

Your turn.

 

[Alan L.]

I think it all depends on how big the thing is. I do know that when you start making those turns that are sharper than 90 degrees (60 degrees in an equilateral triangle) you can't make the turn with the nice neat point you can draw on a piece of paper. What you end up with is each side sort of concave and it gets worse and worse as you go. How to avoid it? I don't know. But turning all the way around - going back and forth takes longer and wastes time too. I'd say the only way to figure out which is faster is trial and error.

Alan L., TX

 

[GrantMO]

The area is not too big, about 200 feet and more or less equilateral. I've tried the parallel to one side method and going around and around the edge. The former has the problem of turning alot and the latter has the problem of the corners getting missed or bulging out as you come back in on the side. Then you have to go back and clean up the corners. Another way I've found to mow around the edges is after a few laps mow straight down a side and do a left hand loop turn (my mower discharges left) and come back down the next side. Any way you do it there are going to be plenty of turns.

<P ID="edit"><FONT SIZE=-1>Edited by grantmo on 07/01/01 00:18 AM (server time).</FONT></P>

 

[GrantMO]

Ron,

The HST and bigger mower is an excellent solution. I'll take it up with the boss. /w3tcompact/icons/smile.gif

 

[patrickg]

Excuse, please. To drive in the shape of an equilateral tri-whatsit requires making 120 degree turns, three times. Note that 3X120=360.

 

[patrickg]

How about mowin' to the middle and driving in an ever expanding spiral utill you get within a cut width of a fence, then folow the fence for the last circuit. If it is "nearly equilateral" and small as you said, then you should be able to finish in one possibly two circuits of the triangle. You will not waste as much swath area in repeated passage over previously cut areas. Angular turns (or the approximation most closely alowed in tractor manuvering) are by their nature innefficient. Innefficiency in the other "vastly inferior" solutions offered is predicated on the fact that tractors don't just stop and neatly rotate to make a turn but have radii of curvature to their turns which lead to imperfections that your consumate driving skill alone cannot compensate for. Stated another way tractor paths aren't analogs of point and line geometry, as they have finite widths, turning radii, and if using a mower not centered on the turning center of the tractor there are other positioning issues as well. Having spent, a few times, large chunks of an afternoon in a 45 degree bank flying in a tight spiral, I can assure you that driving in a spiral (in only 2D) will not destroy your ability to walk upright as even I survived my 3D version multiple times. (They call it gliding/soaring. We had to use human element range extenders while you have the convenience of the shelter of a large back tire.)

Patrick

 

[glennmac]

Patrick,

Yes. Using RPM's diagram, the perimeter pattern would require 14 120 degree turns, and the parallel pattern would require 14 180 degree turns. So, which is more "efficient".

As a practical matter, even if you wanted to drive the parallel pattern, I think you would have first to make at least 2 or 3 perimeter runs so that you would have enough room to turn make the 180 degree turns for the parallel pattern--unless you can do 180 degree turns on your neighbor's lawn. So that would force you into a pattern compromise.

 

[RPM]

Out with the calculator again (I must be warped - I enjoy these puzzles way too much). Course this is all for an equilateral triangle - but I'm pretty sure it's good for any.

If you look at the attached diagram I've added some notation.

'A' is the height of each of the three smaller triangles when mowing in a series of triangles (see previous note).

'B' is the height of the one big triangle when mowing parallel strips.

One turn is required at the end of each strip that is mowed - so we need to count the number of strips for each case.

If we say the the width of the mower is 'w' then:

For the big triangle, the total number of turns will be N1 = B/w

For the three smaller triangles the total number of turns will be N2 = 3A/w

Now for some quick math - let's say the length of each side of the big triangle is 'L'. From the diagram then ...

A = (1/2) x L x sin(30) = 0.25 L

B = (1/2) x L x sin(60) = 0.43 L

So ...

For the triangle approach:

N1 = 3 A / w = 3 x 0.25L / w = 0.75 L/w

For the parallel strip approach (one big triangle)

N2 = B / w = 0.43 L/w

So there are definitely less turns to make for the big triangle. In fact, for the triangle within a triangle approach you have to make 0.75 / 0.43 = 1.75 times more turns.

In my experience (I have a triangular bit of lawn to one side of the drive in the front yard) - on my BX it takes pretty much the same time to turn through 120 degrees as it does 180 degrees. There may be around a 1s difference - but it's small.

My current thinking (and goodness knows I might still have missed something) is therefore that the parallel strips along the longest side approach wins the day.

P.S. Now don't go messing with the math by putting trees, children's toys or mailboxes in the mix!!!

 

[RPM]

I should note that my original diagram was not to scale - stripes were different widths - not meant to be misleading but I couldn't get them to come out the same in MS PowerPoint. So counting the turns won't help - sorry about that - see my following post with a bit of trigonometry for a better answer.

It must be a part of the Microsoft conspiracy!!!