Efficient Computation of Complex Polygon Areas?

In summary, the person is looking for a general formula to approximate the enclosed area of a polygon, whether it is complex or not. They are also looking for advice on how to compute the area of a complex quadrilateral.
  • #1
SonyAD
68
0
Hello. Nice to be here.

If I may, I would like to inquire about the enclosed area of complex polygons. Is there a general formula that will work for these and reduce/cancel out partly for simple/non self-intersecting polygons for a correct enclosed area of theirs as well?

I need to compute the area of a hexagon that may or may not be complex, depending on the circumstances. I don't know whether it is going to be complex or not beforehand and I wouldn't want to test for that because I want a simple and fast algebraic solution of the 'inside in' enclosed area of the polygon. I only really care about the 'inside in' enclosed areas but anything would work better than the 'inside in' - 'inside out' areas I can work out by triangulating the polygon.

For example, when triangulating, there is one possible case when the hexagon degenerates into something akin to the symbol for radiation and then the core will have positive area and the leafs negative areas. I don't know whether the 'inside out' regions will overlap between themselves or the 'inside in' region. If they overlap with the 'inside in' region then they seem to also have positive area, which stacks.

I tried finding a formula for the area of a quadrilateral that will work regardless whether it is complex in the hope of breaking the hexagon up into two quadrilaterals and computing its true area as the sum of theirs.

No luck.

I could use some advice.

This is the last or penultimate roadblock to something potentially very big. :)
 
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  • #2
Ok, so this is what I mean:

[PLAIN]http://img337.imageshack.us/img337/2050/hexagon2.png [Broken]

By triangulating the hexagon I can compute its area:

[tex]A_{123} + A_{134} + A_{145} + A_{156}[/tex]
=
[tex]\frac{x_{3}(y_{2}-y_{4}) +
x_{4}(y_{3}-y_{5}) +
x_{5}(y_{4}-y_{6}) +
x_{6}(y_{5}-y_{1}) +
x_{1}(y_{6}-y_{2}) +
x_{2}(y_{1}-y_{3})}{2}[/tex]

But for the complex hexagon the area yielded is the green minus the red. I need the green area...

What can I do?

Should I try to find a formula that works out the area of simple and complex quadrilaterals, if there is such a thing, and break the hexagon into 2 quadrilaterals then add their areas?
 
Last edited by a moderator:
  • #3
Ok then.

[PLAIN]http://img444.imageshack.us/img444/6765/patrulater.png [Broken]

Ok, so it occurred to me that I can compute the area of any quadrilateral in this fashion:

[tex]\frac{A_{512} + A_{523} + A_{534} + A_{541} + | A_{512} | + | A_{523} | + | A_{534} | + | A_{541} | }{2}[/tex]

I don't know whether this can be used to quadulate more complicated self-intersecting polygons for purposes of computing the area.

This will nicely cancel out the red patches and will also work for complex quads.

Which, alas, brings me to my next question:

How would I go about computing P5 (5.x & 5.y) ?

As can be discerned from the figure, I can't just very well barge on ahead and compute the intersection [P1,P3] ∩ [P2,P4].

I need to compute 5.x and 5.y as a function of all the 4 vertices themselves, not anyone intersection of sides and/or diagonals.

Any suggestions will be appreciated. Thanks.
 
Last edited by a moderator:

1. What is a complex polygon?

A complex polygon is a geometric shape made up of multiple straight sides and angles. It can have any number of sides, as long as all of its angles are less than 180 degrees.

2. How is the area of a complex polygon calculated?

The area of a complex polygon can be calculated by dividing it into smaller, simpler shapes (such as triangles) and finding the sum of their areas. This can be done using various mathematical formulas, depending on the specific shape of the complex polygon.

3. Can the area of a complex polygon be negative?

No, the area of a complex polygon cannot be negative. It is always a positive value, representing the amount of space enclosed by the polygon.

4. How does the number of sides affect the area of a complex polygon?

The number of sides in a complex polygon does not have a direct effect on its area. However, as the number of sides increases, the area tends to become more accurate as the polygon approaches a regular shape with equal sides and angles.

5. What are some real-world applications of calculating the area of complex polygons?

Calculating the area of complex polygons is useful in many fields, such as architecture, city planning, and land surveying. It is also used in computer graphics to determine the size and shape of objects in a digital environment.

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