Thread: Math trivia
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Old 03-07-2009, 04:06 AM
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Goliath Goliath is offline
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Default Re: Math trivia

Suppose that the vertices of our polygon--when traversed counter-clockwise around the polygon--have coordinates (x1, y1), (x2, y2), ...., (xn, yn) for some positive integer n which is at least 3 (so if n=3 you have a triangle, if n=4 a quadrilateral, etc).

As a beautiful consequence of Green's Theorem, areas that are enclosed by a curve (on a plane) can be expressed in terms of the line integral around the curve. So, it turns out that the area of our polygon above can be written as:


(Sorry for the crappy subscripts.)

What's also kinda neat is that, as a corollary, the quantity above is never negative (it's an area!), which isn't immediately obvious, since it contains a lot of sums and differences.
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But (03-09-2009), ceptimus (03-07-2009), Dragar (03-07-2009)
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