Area of Irregular Polygon

-hh hh at livecode.org
Wed Nov 11 19:03:17 EST 2015


> James H. wrote:
> Multiplying this out you get:
> 
> x(i)*y(i+1) - x(i+1)y(i) +      [x(i+1)* y(i+1) - x(i)* y(i)]
> 
> So THE SAME EXPRESSION as in the centroid method EXCEPT for the added term in square brackets. So, WHY THE  DIFFERENCE?
> 
> In calculating this sum all of the intermediate terms in the sum cancel out, leaving just the end terms:  x(n)y(n) - x(1)* y(1)
> But if the figure is closed, they too cancel each other. 

Very clear explanation (it's in fact a proof) of the 'shoelace'-formula.

> James H. wrote:
> The center of mass bears no relation to the mass, and the centroid of a closed curved bears no relation to the area.

The connection between the centroid and the Area is purely 'scottish'.
We need the terms of the area-sum for centroid's computation and have by that the area computed as a by-product. Because we are canny as the Scots (and the people from Southern Germany and Switzerland) we don't waste it, but return it too.

If one takes a simple flat map (no relief), a centroid is good as a *first* approximation where to have a central storage that is the origin of catering goods for a certain region.
Likewise one could use the centroid as an approximate good place for a time-and-cost sparing residence in a state or country.





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