Is there a formal definition of the area of a shape?
To give an idea of what I'm looking for:
Given a prepositional function, p(x, y), where x and y are real, which is true when x and y are on a shape, define the area of the shape for all p?
Formal mathematical definition of area
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Re: Formal mathematical definition of area
That would be the integral of the prepositional function, taken to have value 0 outside the shape, and 1 inside, over the plane.
Of course, no one uses this in practice. Far more useful would be to have a function that gives the border of the shape. If the boundary (which is defined as the set of points where every neighborhood[0] includes both a point inside the shape and a point outside the shape, or alternately, the set of points in the closure[1] of the shape but not its interior[2]) can be expressed as two functions y1(x) and y2(x) over some domain from a to b (and note that many shapes that do not meet this criterion can be broken down into shapes that do) as the integral, from a to b, of the absolute value of the difference of these two functions, with respect to x. Similar constructs can be assembled for polar systems or for threedimensional volumes.
There's also always measure theory. But that's scary and a little bit disconcerting.
Definitions because I don't know how much you know:
[0]: A neighborhood of a point is an open, simple, connected set containing that point
[1]: The closure of a shape is the set of all points in that shape as well as any points for which every neighborhood of that point contains a point in the shape
[2]: The interior of a shape is the set of all points in that shape for which every neighborhood of that point contains no points that are not in that shape
Of course, no one uses this in practice. Far more useful would be to have a function that gives the border of the shape. If the boundary (which is defined as the set of points where every neighborhood[0] includes both a point inside the shape and a point outside the shape, or alternately, the set of points in the closure[1] of the shape but not its interior[2]) can be expressed as two functions y1(x) and y2(x) over some domain from a to b (and note that many shapes that do not meet this criterion can be broken down into shapes that do) as the integral, from a to b, of the absolute value of the difference of these two functions, with respect to x. Similar constructs can be assembled for polar systems or for threedimensional volumes.
There's also always measure theory. But that's scary and a little bit disconcerting.
Definitions because I don't know how much you know:
[0]: A neighborhood of a point is an open, simple, connected set containing that point
[1]: The closure of a shape is the set of all points in that shape as well as any points for which every neighborhood of that point contains a point in the shape
[2]: The interior of a shape is the set of all points in that shape for which every neighborhood of that point contains no points that are not in that shape
Last edited by ST47 on Wed May 11, 2011 6:44 pm UTC, edited 2 times in total.

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Re: Formal mathematical definition of area
For most reasonable shapes, the Riemann integral works fine. If you are interested in a more general definition, I would suggest reading up on measure theory. Wikipedia has pretty good articles on both topics.
Re: Formal mathematical definition of area
ST47 wrote:Of course, no one uses this in practice.
Actually this is a powerful technique. Coordinate transformations and fubini's theorem allow you to calculate the integrals easily.
The most general theory that allows you to define areas/volumes are volume forms on differentiable manifolds.
Integrals over differentiable manifolds are a natural way to express many volumes. Stokes' theorem allows you to calculate many such integrals in lower dimensions.
Measure theory is required to define the Lebesgue integral. The Riemann integral sucks, especially when you take integrals over unbound subsets or if you want to integrate more complicated functions.
Re: Formal mathematical definition of area
ST47 wrote:[2]: The interior of a shape is the set of all points in that shape for whicheverysome neighborhood of that point contains no points that are not in that shape
Fixed.
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 Yakk
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Re: Formal mathematical definition of area
A measure mu is a function from some subset of the power set of some set S to the positive reals with the following properties:
1) mu(empty set) = 0
2) mu(a U b) where a and be are disjoint = mu(a) + mu(b)
3) (2) is true for countable disjoint unions, with a sensible definition of the infinite sum.
Countable means "you can assign a unique integer index to each of the elements". This happens not to be true of the real numbers, for example.
We can then build a measure on the reals by starting off with measure of n x m blocks as being size n*m, and jump through some technical hoops to make it "nice". The name for this is "Lebesgue measure" (well, you can get at it a few ways), and given certain subsets of R^n, it returns the ndimensional 'area' of the subset. Note that it doesn't work on every subset of R^n (at least under certain standard axiomizations of the real numbers).
We also use this to define integration.
Now, for less pathological cases, we can use simpler definitions of area, and they work reasonably well.
1) mu(empty set) = 0
2) mu(a U b) where a and be are disjoint = mu(a) + mu(b)
3) (2) is true for countable disjoint unions, with a sensible definition of the infinite sum.
Countable means "you can assign a unique integer index to each of the elements". This happens not to be true of the real numbers, for example.
We can then build a measure on the reals by starting off with measure of n x m blocks as being size n*m, and jump through some technical hoops to make it "nice". The name for this is "Lebesgue measure" (well, you can get at it a few ways), and given certain subsets of R^n, it returns the ndimensional 'area' of the subset. Note that it doesn't work on every subset of R^n (at least under certain standard axiomizations of the real numbers).
We also use this to define integration.
Now, for less pathological cases, we can use simpler definitions of area, and they work reasonably well.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Re: Formal mathematical definition of area
Yakk wrote:Note that it doesn't work on every subset of R^n (at least under certain standard axiomizations of the real numbers).
Also note that it only allows you to determine the ndimensional volume in R^n. There are measures that allow you to measure the volume of sets in arbitrary dimensions: http://en.wikipedia.org/wiki/Hausdorff_measure
 MartianInvader
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Re: Formal mathematical definition of area
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!
Re: Formal mathematical definition of area
Thanks, to be honest I'm hardly understanding anything of measure theory [math]my knowledge of set theory = \{\cap, \cup\}[/math] but now I know where to look. I really want to learn something new in math, but I never have the background for something new.
 imatrendytotebag
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Re: Formal mathematical definition of area
Intuitively, the idea of measure theory (lets do 2dimensional are right now) is pretty straightforward.
1: We want a rectangle with width a and length b to have area ab.
2: If we know the area of region 1 and the area of region 2, and the regions don't overlap, then the area of region 1 and 2 is just the sum of the areas of region 1 and region 2.
3: In fact, if we know the areas of region 1, region 2, ... region n, ... etc. (potentially an infinite series of regions) and ALL the regions don't overlap, then the combination of all the regions should have an area equal to the sum of region 1 + region 2 + ... + region n + ...
4: If region 1 is contained inside region 2, region 1 has a smaller area.
These all are very basic ideas, and they lead to the ability to measure the area of many subsets on the plane.
In order to really attack this concept mathematically though, you will probably need to get a little more comfortable with set theory.
1: We want a rectangle with width a and length b to have area ab.
2: If we know the area of region 1 and the area of region 2, and the regions don't overlap, then the area of region 1 and 2 is just the sum of the areas of region 1 and region 2.
3: In fact, if we know the areas of region 1, region 2, ... region n, ... etc. (potentially an infinite series of regions) and ALL the regions don't overlap, then the combination of all the regions should have an area equal to the sum of region 1 + region 2 + ... + region n + ...
4: If region 1 is contained inside region 2, region 1 has a smaller area.
These all are very basic ideas, and they lead to the ability to measure the area of many subsets on the plane.
In order to really attack this concept mathematically though, you will probably need to get a little more comfortable with set theory.
Hey baby, I'm proving love at nth sight by induction and you're my base case.
 Yakk
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Re: Formal mathematical definition of area
Math is a deep subject, and if you randomly select something in it, you'll end up . What is your background?
Finding things that are "shallow" to you isn't hard  but expecting a random thing to be "shallow" isn't that reasonable.
Finding things that are "shallow" to you isn't hard  but expecting a random thing to be "shallow" isn't that reasonable.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
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