From:	Erik Max Francis <max AT alcyone DOT com>
Newsgroups:	comp.os.msdos.djgpp
Subject:	Re: Searching for a very specific algorithm
Date:	Thu, 29 May 1997 21:41:14 -0700
Organization:	Alcyone Systems
Message-ID:	<338E5A6A.4D22B301@alcyone.com>
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To:	djgpp AT delorie DOT com
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Alessandro Moure wrote:

>         Hi all. I am developing a data manipulation tool kit and I am
> experiencing a hard time trying to find the area inside a closed,
> generic region. I have the points of the border line of the region
> stored in a vector. Does any body knows how to do it?

The most general form would be a corrolary of Green's theorem, which
allows you to calculate the area (double integral over a region) with a
line integral:

    2 dA = x dy - y dx.

and integrate over the curve.  If your curve is parameterized with
parameter t, you can write this as 

    2 dA = (x dy/dt - y dx/dt) dt, 

and integrate.  Since in general you won't be able to find dx/dt or dy/dt,
and possibly won't be able to antidifferentiate the right side of the
equation, you can change this to deltas and write 

    deltaA = (1/2) (x deltay/deltat - y deltax/deltat) deltat, 

and add up the deltaA's from t = 0 to t = t' (the end of the closed
curve), and compute deltax == x(t + deltat) - x(t) and deltay == y(t +
deltat) - y(t).

-- 
       Erik Max Francis, &tSftDotIotE / email / max AT alcyone DOT com
                     Alcyone Systems /   web / http://www.alcyone.com/max/
San Jose, California, United States /  icbm / 37 20 07 N  121 53 38 W
                                   \
     "Covenants without the sword / are but words."
                                 / Camden

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