November 13th, 2013, 01:13 PM  #1 
Newbie Joined: Nov 2013 Posts: 4 Thanks: 0  center of mass
Hi there, I want to find the center of mass of any shape. I know the area of the shape and the coordinates of each point at the borders of the shape. The latter is also called a chain code, it is used in image processing. Example: Code: x x x x x x x x x x x x x x x x Area = 16 Coordinates of shape (starting from top left): 1,4;1,3;1,2;1,1;2,1;3,1;4,1;4,2;4,3;4,4;3,4;2,4 Code: x x x x x x x x x x x x x x x x x x x x Area = 20 Coordinates of shape (starting from top left): 1,3;1,2;2,2;2,1;3,1;4,1;5,1;5,2;6,2;6,3;5,3;5,4;4,4;3,4;2,4;2,3 
November 13th, 2013, 03:42 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,155 Thanks: 1422 
How do you know the area?

November 14th, 2013, 12:27 AM  #3 
Newbie Joined: Nov 2013 Posts: 4 Thanks: 0  Re: center of mass
I know the area by determining the direction of the chain's coordinate. If it goes down, substract the value of the xcoordinate, if it goes up, add the value of the xcoordinate + 1. E.g. for the first example the summation will be: 1111+5+5+5+5 = 16. The algorithm is actually little bit more complex, because of the chain going around corners, since it is not really related to my question (how to find the center) I will not describe it in detail. You may assume the area is known and correct. If I did post this in the wrong category, please don't hesitate to move it, I'm new here. 
November 14th, 2013, 02:09 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 18,155 Thanks: 1422 
Does your area calculation work for a shape that is not convex? Also, I doubt that it works adequately for a shape that includes (or excludes) a long, narrowing spike. If your shape can be decomposed into nonoverlapping rectangles, triangles, circles and certain other basic shapes for which the area and C of G are known, you can then find the overall C of G fairly easily, but making sure the components are nonoverlapping might be tricky (the sum of the areas of the components must be the same as the overall area). You could test your area calculation method on the shape below. [attachment=0:2u3p1ali]Areas.gif[/attachment:2u3p1ali] 
November 14th, 2013, 04:50 AM  #5 
Newbie Joined: Nov 2013 Posts: 4 Thanks: 0  Re: center of mass
Because the application is grid based (pixels in image processing) I have changed your shape to the one attached and here is the ouput. Code: area =37 chain length =32 x[0] =1 y[0] =1 x[1] =1 y[1] =2 x[2] =1 y[2] =3 x[3] =1 y[3] =4 x[4] =1 y[4] =5 x[5] =2 y[5] =5 x[6] =2 y[6] =4 x[7] =3 y[7] =4 x[8] =4 y[8] =4 x[9] =5 y[9] =4 x[10] =5 y[10] =3 x[11] =6 y[11] =3 x[12] =7 y[12] =3 x[13] =7 y[13] =2 x[14] =8 y[14] =2 x[15] =9 y[15] =2 x[16] =10 y[16] =2 x[17] =10 y[17] =1 x[18] =11 y[18] =1 x[19] =12 y[19] =1 x[20] =13 y[20] =1 x[21] =12 y[21] =1 x[22] =11 y[22] =1 x[23] =10 y[23] =1 x[24] =9 y[24] =1 x[25] =8 y[25] =1 x[26] =7 y[26] =1 x[27] =6 y[27] =1 x[28] =5 y[28] =1 x[29] =4 y[29] =1 x[30] =3 y[30] =1 x[31] =2 y[31] =1 I don't see how I can decompose my shape in the largest possible basic shapes. I can decompose it in rectangles of width 1 (basically the way I calculate the area), but what to do with that? 
November 14th, 2013, 06:24 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 18,155 Thanks: 1422  Re: center of mass
I'm a bit confused. You originally referred to "any shape", not just a shape made up of squares. If you deem the shape to be made up of nonoverlapping squares (or rectangles) and you can list them and their sizes, you can find the C of G quite easily (though narrow spikes would presumably need special rules). Its position might not correspond exactly with a pixel position. What is the area if one excludes everything above the long blue diagonal line? Is it the sum of the areas of the outlined components found by the same method? 
November 17th, 2013, 12:40 PM  #7  
Newbie Joined: Nov 2013 Posts: 4 Thanks: 0  Re: center of mass
Thanks skipjack, this Quote:
 

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