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Re: Filled area inside a closed curve

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I do this post because I am uncertain that you got my previous post.
Thanks a lot for your response on my question.
It seems that my question is a bit incomplete so I clarify it more.
Suppose a sea that contains and island. The island may have some lakes
and each lake may have some islands. So lakes and islands are nested.
Lakes are disjoint and islands inside each lake are also disjoint.
I can only measure the area size of islands and lakes. For example if
an island have a lake inside I cannot measure the covered area by soil
in the island and I subtract the area size of lake from island to find
the area covered by soil inside island.
Please consider the figure
http://imageshack.us/photo/my-images/13/unledtmn.png/
I cannot measure the areas between islands and lake and I must
calculate them.
I cannot find the area size of partitions you defined (D = {c in C:
not exists u in C: u subset c}) . But they are calculable.
I would appreciate if you verify that I explained the problem
correctly and provide me its proof.
Thanks again.




On Aug 9, 9:34 pm, Kaba <k...@nowhere.com> wrote:
> Saeed wrote:
> > Consider a closed curve that contains some holes. I would like to show
> > that the filled area can be calculated by subtracting the area of
> > holes from area of the curve. Is there any theorem that can prove it?
>
> Intuitively
> -----------
>
> Consider the partition the curves induce, and then sum up the measures
> of those regions which form the actual region you wanted to measure.
>
> Formally
> --------
>
> Assume that
>
> area = (Lebesgue) measure in RR^2, and
> subset enclosed in a closed curve = measureable subset of RR^2 with a
> boundary of measure zero.
>
> Then the result is a direct consequence of the additivity of the
> measure.
>
> http://en.wikipedia.org/wiki/Measure_(mathematics)
>
> Let B = {B_1, ..., B_k}, where k in NN, and B_i subset RR^2.
>
> Let C = {b in B : b = intersect_{i in P(k)} B_i},
> where P(k) is the set of subsets (powerset) of [1, k] subset NN.
> I.e. the set of all intersections of the sets in B.
>
> Let D = {c in C: not exists u in C: u subset c}. Then union D = union B,
> and the interiors of the sets in D are disjoint. Since the boundaries of
> the sets in B were assumed to be of measure zero, so are the boundaries
> of the sets in D, and the additivity of measure applies to these sets
> (although they are not necessarily disjoint).
>
> Associate each set d_i in D with either 1 or the 0 by the function
> s : NN --> RR, denoting whether the set is a hole or not.
>
> Then the total measure of the "1-sets" in D is given by
>
> M_1 = m(union_{i, s(i) = 1} d_i)
> = sum_{i, s(i) = 1} m(d_i)        (by additivity of measure).
>
> Similarly the total measure of the "0-sets" in D is given by
>
> M_0 = sum_{i, s(i) = 0} m(d_i)
>
> The measure M = m(union B), is
>
> M = m(union B)
> = m(union D)          (since union D = union B)
> = M_0 + M_1           (by additivity of measure),
>
> which leads to
>
> M_1 = M - M_0,
>
> i.e. "the total measure of the 1-sets is the total measure of all sets
> in D minus the total measure of the 0-sets".
>
> --http://kaba.hilvi.org

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Filled area inside a closed curve Saeed <saeedsedighian@gmail.com> - 2011-08-09 01:03 -0700
  Re: Filled area inside a closed curve Kaba <kaba@nowhere.com> - 2011-08-09 20:34 +0300
    Re: Filled area inside a closed curve Saeed <saeedsedighian@gmail.com> - 2011-08-11 05:30 -0700
      Re: Filled area inside a closed curve Kaba <kaba@nowhere.com> - 2011-08-11 18:12 +0300
        Re: Filled area inside a closed curve Saeed <saeedsedighian@gmail.com> - 2011-08-12 04:51 -0700
          Re: Filled area inside a closed curve Kaba <kaba@nowhere.com> - 2011-08-12 16:50 +0300
            Re: Filled area inside a closed curve Saeed <saeedsedighian@gmail.com> - 2011-08-12 22:17 -0700

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