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All of the questions I found are about literally covering "What is the most effective way to cover a circular pot with a square lid etc..." Here's an example to show what I mean: I have a circular farm of 10,000 sq m in area. I have a crop duster that can cover a ten meter wide area (10 meter wingspan) while traveling in a line. How would I determine the most efficient flight path to crop dust my farm while using as little fuel as possible? Should I spiral inwards? Fill the outer circle, then move in for concentric, separate circles? Make a sun-beam-like motion back and forth from the center? This could also be applied to patroling a security area, sweeping a circular room, etc. Would it always be the same, regardless of wingspan and area size?

Walt
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  • Tough to say without knowing what constitutes efficient flight... a circular field with area $1000m^2$ has a radius of about $17.84$, so you could simply fly in a tight circle (so that your wingtip traces the circumference), and then double back to cover the remaining "inner circle" in one pass by flying in another tight circle (so that your wingtip remains centred on the centre of the field). There will be an overlap in the shape of an annulus with an outer radius of $10m$ and an inner radius of $7.84m$ for a total overlap of $\pi(10^2-7.84^2)m^2$ – H. sapiens rex Feb 25 '24 at 23:35
  • This is an overlap of about $120.99m^2$. Relative to the size of your field, it's an overlap of about $12.099%$. Perhaps another strategy will do better – H. sapiens rex Feb 25 '24 at 23:38
  • @H.sapiensrex Ok, I suppose a 17.84 meter radius and a 10m wingspan does make it kind of obvious, but I'm curious as to whether concentric circles, spiraling, or whatever else are better for significantly larger areas (like 100 meter radius and onwards). Also, in this context, I just mean efficient flight as least distance flown, though I suppose least overlap works as well. – Walt Feb 26 '24 at 01:27
  • Well, circles have the lowest perimeter-to-area ratio: imagine a circle and a square, both of area $1m^2$. Then the circle has perimeter $2\sqrt{\pi}$ while the square has a perimeter of $4$, which is longer. Generally, the more you deviate from a circular path for a given area, the greater the perimeter/distance you'll end up travelling. So if your criterion for efficient crop dusting is that which minimizes the distance travelled, then you can't do better than concentric circles (which is what I suggested) – H. sapiens rex Feb 26 '24 at 02:29
  • @H.sapiensrex, ok, thank you! – Walt Feb 26 '24 at 14:27

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