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The COM of a point is the point itself. With another point you can construct a line. The COM of a line is 1/2 the distance between the COM the old point and the new measured form the former, with another point you can construct a triangle.

The COM of the triangle is located at one third the distance between the COM (midpoint) of the old line and the new point, measured from the COM of the old line.

I am trying to find out if the following statement is true: Given an (n-1)-gon, combined with a point, can be extended to a n-gon, the COM of the created n-gon is 1/n *L away from the COM of the original (n-1)-gon lying on the line joining the n-th point with the former, where L is the distance between the (n-1)-gon's COM and the n-th point.

for example, consider a triangle ABC, and a point D. let the COM of ABC be P. D, together with ABC can be used to construct the tetragon ABDC. Should the COM of ABDC lie on PD, at a distance 1/4 * PD from P? For a uniform 4-gon, it is easy to show that the statement is true. For any general 3-gon, it holds. What happens for a general n and an non-uniform polygon?

Sean
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1 Answers1

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COM means Centre of mass, right?

If the new point adds a new dimension (e.g. point to line to triangle to tetrahedron to ...) then the result is true.

In two dimensions, it is not true. For a counterexample, imagine D very close to AB. It adds no mass, so the COM does not move.

apt1002
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  • Yes, Center of Mass,

    However, I am able to proove it for a regular 4 and 5-gon, in 2 dimensions, and for the statement about adding a new dimension, is there a proof i can look at?

     – Sean Nov 14 '13 at 17:33
  • I am surprised it works for a regular 5-gon! – apt1002 Nov 14 '13 at 17:43
  • To find the COM of a tetrahedron, we can slice it up parallel to the base. Define a coordinate $x$ which is 1 at the base and 0 at the tip. Let $A$ be the area of the base. Then the area of slice $x$ is $Ax^2$ (similar triangles). So the COM is at $$\frac {\int_0^1 xAx^2 ~dx}{\int_0^1 Ax^2 ~dx}$$ The result is $\frac34$. In other words, the COM is $\frac14$ of the way from the base to the tip. I hope it is clear how to generalise to any number of dimensions. – apt1002 Nov 14 '13 at 17:55
  • yes, thank you! – Sean Nov 14 '13 at 18:57