Thread: Math trivia
View Single Post
  #32  
Old 03-07-2009, 04:06 AM
Goliath's Avatar
Goliath Goliath is offline
select custom_user_title from user_info where username='Goliath';
 
Join Date: Jul 2004
Location: Kansas City, MO
Gender: Male
Posts: MMDCCVII
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:

(1/2)*[(x2y1-x1y2)+(x3y2-x2y3)+(x4y3-x3y4)+...+(x1yn-xny1)].

(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.
__________________
Cleanliness is next to godliness.
Godliness is next to impossible.
Therefore, cleanliness is next to impossible.
Reply With Quote
Thanks, from:
But (03-09-2009), ceptimus (03-07-2009), Dragar (03-07-2009)
 
Page generated in 0.17422 seconds with 11 queries