Imagine a regular grid of circles:
ooooo
ooooo
ooooo
The radius of the circles is variable, and thus the circles may
overlap or they may not (but all the circles must have the same
radius). If the circles are not overlapping, the average area of a
space, as defined by any space bounded by lines, is simply Pi.r^2.
However, once the circles start to overlap, some of the spaces are the
area of the circle minus the "bites" taken out of them by the
overlapping area, and some of the spaces are the parts that overlap
with each other. Is there any formula that will give me the average
area of all the spaces? Note that the radius of the circle has no
limit - i.e there is nothing wrong with the boundery of the circle
passing beyond the center of an adjacent circle.
If you can't think of any formula, is there at least any proof for or
against my hypothesis that the graph of the average area of the spaces
will initially go up as the radius of the circles increases, but then
start to go down once the circles start to overlap, and keeps going
down as the radius of the circles increases. Would it make a
difference if the number of circles were finite or infinite? If it
were
finite, the graph would actually have to start going up, probably once
the radius of the circles is larger than the size of the grid. Does it
make a difference if the circles are not aranged in a grid? |
Request for Question Clarification by
josh_g-ga
on
25 Nov 2002 07:12 PST
By "average area of the spaces", do you mean the area covered divided
by the number of circles? That is, the average area covered per
circle?
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Clarification of Question by
bobtherat-ga
on
25 Nov 2002 07:29 PST
No, I defined "spaces" above as the area bounded by any lines. thus,
some spaces might look like circles with "bites" taken out of them,
and some might be the intersection of two or more circles. For
instance, if three equidistant circles are overlapping each other,
then there are seven spaces, one for each of the circles A, B and C
(with a bite taken out of them), one bewteen A and B, one between B
and C, one between A and C, and one in the middle, shaped like a
triangle with curved sides. What is the average size of these spaces.
This answer would almost certainly have to be a function of radius,
distance between the circles(?) and number of circles(?). Or, if that
can't be answered, can you describe and prove the shape of the graph
of average size vs. radius. I'm expecting that it will look sort of
like a x^3 graph - starts out at 0, increases as the radius increases,
decreases once the circles start to over lap, and then starts
increasing again at a certain point. This isn't necessarity the
correct andwer, though.
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Request for Question Clarification by
aceresearcher-ga
on
25 Nov 2002 07:30 PST
Hi bobtherat,
I'm confused! First you say "The radius of the circles is variable",
and then you say "(but all the circles must have the same radius)".
Both cannot be true!
aceresearcher
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Clarification of Question by
bobtherat-ga
on
25 Nov 2002 07:31 PST
Looking at is as the area covered divided by the number of spaces will
probably help, though.
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Clarification of Question by
bobtherat-ga
on
25 Nov 2002 07:33 PST
We posted at the same time. Yes, it can be true, because all of the
circles must have the same radius as each other, but this radius is
not fixed. They could all have a radius of 1 cm or all have a radius
of 2 cm, for instance.
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Request for Question Clarification by
mathtalk-ga
on
25 Nov 2002 09:37 PST
Let's see if I've got it now. You start with small circles centered
on a square grid. As the common radius of these circles increases,
they first touch and then overlap. Find the average area of those
bounded regions as a function of r.
Do we count area that lies outside of a circle but happens to be
bounded by the exteriors of more than one circle?
Would an approximation, valid as the number of grid points increases
without limit be acceptable? (Your illustration shows a 3x5
arrangement of circles. I have in mind an NxN arrangement as N tends
to infinity.)
regards, mathtalk-ga
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Request for Question Clarification by
bobby_d-ga
on
25 Nov 2002 13:13 PST
Woah - thanks josh-g, nearly answered this one without realising what
"spaces" meant!
Here's how it needs to be approached, in my mind...
Firstly, we need to find a formula relating x (the radius) and a
variable parameter p (the amount of spaces), according to a set
distance between each adjacent circles' centres (say, a) and the
amount of circles (n)
From here, we then need to find a relation between x and the total
area covered (which wouldn't be too easy, and would also include n and
a, and x). Let's call total area covered "b". Therefore, y would
equal b (which includes x and a and n) divided by p (which includes x
and a and n).
Now I am very confused - I should really stay away from these
questions.
But what we can say is that if n is infinite (ie infinite amount of
circles), therefore there is an infinite amount of space. As soon as
the radius of the circles is root 2 times a (a being the distance
between adjacent circles' centres), an infinite amount of space would
have been covered, and no more space will be covered as the radius
expands.
Am I right?
I hope this adds to whatever future discussion on this question.
Thanks,
Bobby_d
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Clarification of Question by
bobtherat-ga
on
25 Nov 2002 19:36 PST
This has gone on a bit too long for it to still be useful for me, but
I'm interested and so will keep it open. Also because of the interest
you researches have shown.
mathtalk - yes to the fist part, you've understood my question, but
I'm actually more interested in the problem with a finite number of
circles (this all has to do with neurobiology - areas in a cortical
map and such). I don't think that I made it clear, so I'll accept an
answer that uses an infinite number of circles.
I also have a big modifier, which I didn't put in before, so again
I'll accept an answer without using it, but I prefer it if it were
used. It might even make the problem easier:
The circles have to be thought of as being bounded in a square:
------------
|oooooooooo|
|oooooooooo|
|oooooooooo|
|oooooooooo|
|oooooooooo|
------------
The only area that is relevent is that within the square. Thus, as
long as none of the circles overlap the bounderies of the square, the
problem is identical. However, once they do, any space outside of the
square is ignored. Thus, if we imagine the circles all being so large
that we cannot even see their bounderies on the diagram above (i.e.
they are larger than the square), then there is only one space in the
diagram, and that is the size of the square. This puts an upper limit
on the size of the spaces.
Phew! This is far more complicated to explain than it was in my head.
Just so researchers know, so that they don't waste time on it, I'm
thinking of closing the question some time tomorrow morning if there
is no answer, as I am all ready past the time that it is useful for
me.
Thanks!
bobtherat
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Clarification of Question by
bobtherat-ga
on
25 Nov 2002 23:20 PST
Last thing. This will make it easier, I promise! :)
Look at my comment number two below. I've realised that the only time
I really care about this is when there is no "white space" in the
square, i.e. there are no points that are in the square that are not
in at least one circle. Thus, if you can find out the pattern of how
the number of spaces grows, you can work out the problem (see the end
of comment number 1). The end might be in sight.
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