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Topic: How many Birds (Read 12159 times) |
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rloginunix
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Re: How many Birds
« Reply #25 on: Mar 17th, 2014, 4:41pm » |
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I see your point.
I guess I didn't put down explicitly what was always on my mind, keeping it to myself instead - my interpretation of the verb or concept of leadership. In my mind I treated it as leadership of the flock as a whole. So my definition should've been "A bird is a leader of a flock as a whole ...". My bad.
You're proposing a much more narrower interpretation. Doesn't it limit the scope of your definition to just two birds? What does it mean then to say that "two" birds are leading one since there's no such definition. What am I missing? Also, how would one geometrically distinguish "A leads B" from "B leads A". Are you keeping geometry or defining it in terms of algebra alone, like "L" is a binary operator with these properties, etc? Thanks.
Circles are tricky. Velocity vectors will be tangent to a circle. And I can always find a formation where each vector (extended) intersects another FP making it not clear at all where does the flock begin and where does it end.
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rmsgrey
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Re: How many Birds
« Reply #26 on: Mar 18th, 2014, 8:00am » |
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on Mar 17th, 2014, 4:41pm, rloginunix wrote:I see your point.
I guess I didn't put down explicitly what was always on my mind, keeping it to myself instead - my interpretation of the verb or concept of leadership. In my mind I treated it as leadership of the flock as a whole. So my definition should've been "A bird is a leader of a flock as a whole ...". My bad.
You're proposing a much more narrower interpretation. Doesn't it limit the scope of your definition to just two birds? What does it mean then to say that "two" birds are leading one since there's no such definition. What am I missing? Also, how would one geometrically distinguish "A leads B" from "B leads A". Are you keeping geometry or defining it in terms of algebra alone, like "L" is a binary operator with these properties, etc? Thanks.
Circles are tricky. Velocity vectors will be tangent to a circle. And I can always find a formation where each vector (extended) intersects another FP making it not clear at all where does the flock begin and where does it end.
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I would interpret "two birds lead one" as "there are two birds, A and B, and a third bird, C, such that A leads C and B leads C" (equivalently, "there is a bird, for which there are another two birds, each of which is leading it")
I've (deliberately) not fully specified the "leading" relationship - in particular, I've left out how you'd decide which of the three possibilities applies to any given pair of birds. That's partly because there are several plausible schemes - I'd probably settle for "A is leading B if the direction of the vector BA is within a certain angle, theta, of both the direction of A's motion and the direction of B's motion". As a rough idea, I'd set theta as 60 degrees, but that's open to debate. The obvious hole in this definition is the lack of any proximity criterion - if there are two birds thousands of miles apart who both happen to be flying in the right direction, is one really leading the other? The other plausible improvement would be some idea of correlation of motion over time - though that's debatable.
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rloginunix
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Re: How many Birds
« Reply #27 on: Mar 19th, 2014, 7:06pm » |
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By carefully delineating the boundaries of my definition - 2D, constant velocity, straight line - I purposefully avoided going into flux, flow, gradient, curl, divergence and all other things vector calculus which I studied more than two decades ago (relative to the time stamp of this message) but never touched since so I can't be immediately productive there. But that's where this issue should really land, I think, in its most generic case since flying birds and vector calculus have common core: items moving through a surface over time.
My thinking was: "is it possible to create such a formation of birds that will satisfy the problem statement but will amount to more or less than a 3-bird flock as a whole".
I would define my greedy flock thus. Spill N points at random over a plane. Drive nails into each point. Stretch a rubber band a lot, pull it over the nails and then let it go. Smallest perimeter convex polygon. In 3D: throw N points at random, freeze them, stretch a rubbery ball with a hole in it over the points, let go, cover the hole. Smallest surface area convex polyhedron.
But in any case I'm still staying in a rigid formation and the definitions apply to all the birds in a flock.
on Mar 18th, 2014, 8:00am, rmsgrey wrote:| would interpret "two birds lead one" as "there are two birds, A and B, and a third bird, C, such that A leads C and B leads C" (equivalently, "there is a bird, for which there are another two birds, each of which is leading it"). |
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Playing devil's advocate here. While we know what's going on between A and C and between B and C we are not really sure what's going on between A and B themselves. Is it important? Not sure.
From your definition it follows (algebraically):
A L C && B L C
Conversly, one bird leads two:
C L A && C L B
Do a simple substitution: A = C', C' = C and we get: one bird leads itself and the other bird behind it, so there are just 2 birds here.
In the following infinite two, one, two, one, two ... formation flying left to right:
* * * *
* * *
* * * *
your pair-based definition stands: A leads C and B leads C, while C leads D and C leads E and so on ad infinitum. How can this be prevented?
Lastly, a question about geometry. I assume that you assume that vectors A and B are not parallel (something I avoid altogether). But if we fix the angles for an arbitrary formation when A is in front then we can always find a different arrangement when B is in front but the angles are the same. Don't you think angle alone is not enough?
Thanks.
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rmsgrey
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Re: How many Birds
« Reply #28 on: Mar 20th, 2014, 8:33am » |
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If A is in front, then the vector BA is going forward (and the angles are acute); if B is in front, the vector BA is going backward (and the angles are obtuse).
I'm talking about the difference in bearing between the two vectors, not the angles at which the lines representing them would meet if you drew them on the same diagram in the obvious way.
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rloginunix
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Re: How many Birds
« Reply #29 on: Mar 20th, 2014, 9:24am » |
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I see now, makes sense.
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