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Centre of gravity of polygon
By Bj Russ on Tuesday, May 21, 2002 - 05:19 pm:

if I have the coordinates of the vertices of an irregular plane polygon, how do I find the position of its centre of gravity ?

By Graeme Mcrae on Tuesday, May 21, 2002 - 06:53 pm:

Look at
http://astronomy.swin.edu.au/~pbourke/geometry/polyarea/
near the bottom of the page.
--Graeme

By Yatir Halevi on Wednesday, May 22, 2002 - 06:57 pm:

Graeme, why does the formula give us the center of gravity. How did someone derive this formula?


Yatir

By Graeme Mcrae on Wednesday, May 22, 2002 - 08:41 pm:

It works by dividing the polygon into triangles (in a very strange way, which I'll talk about in a minute) finding the centroids (i.e. center of gravity) of all the triangles and then takes the weighted average of these centroids.

Now, for the strange way of dividing the polygon into triangles... First, I'll define "signed area" of a triangle with vertices (x1,y1),(x2,y2),(x3,y 3) as

(1/2)(x1y2-x2y1+x2y 3-x3y2+y1x3-x 1y3)

If the vertices are named counterclockwise, the area is positive, and the area is negative if the vertices are named clockwise.

Let P1, P2, ..., Pn be the vertices of the polygon. The strange way we divide this polygon into triangles is by adding an n+1th point, which I'll call O, and making n stangely overlapping triangles as follows:

P1, P2, O
P2, P3, O
...
Pn, P1, O

After a little thought, and some picture-drawing, you'll see that the signed area of these n triangles is equal to the area of the polygon. When you add up the areas of the triangles, you'll see that the coordinates of point "O" cancel out entirely, making it irrelevant where that point is. If point "O" is chosen to be the origin, then the area of each triangle is a function of the coordinates of the other two points, making the sum very easy to calculate.

The centroid of a triangle is simply the average of the three corners, so the formula adds up the centroid multiplied by the area of each triangle, and divides the result by the area of the whole polygon.

The number "6" in the denominator comes from the fact that the centriod of the triangle is the average of three numbers and the area of each triangle has a factor of (1/2) in the formula.

By Yatir Halevi on Monday, May 27, 2002 - 02:56 pm:

Thanks, Graeme!