[Math] How to find the center of mass of a complex 2d shape

algorithmsgeometry

So I'm hoping someone could help me get a solid algorithm for finding the center of mass of 2D shape without density, based of a set of vertices.

Please explain in simple terms that a non-mathematician could understand (I have no clue how the wikipedia equation works) and easily translate into C++ code for my program! =)

Note: if it makes the algorithm any faster, the complex shapes in my program are really just a group of convex polygons which I can access.

Best Answer

Divide your complex polygon into non-overlapping triangles.
The center of mass for each triangle is the simple average of the coordinates of its three corners. (Because this works for an equilateral triangle, and affine transformations preserve centroids and can take any triangle to any other triangle).
Average the centers-of-mass of all the triangles, weighting by the area of each triangle (which you can find as the 2D cross product of two of its sides).

Related Solutions

[Math] Volume of n-dimensional solid w/ n-1 dimensional simplex as a base

I struggled with this one, but it finally became clear. I hope this answer helps someone else along the way.

The comment around the original MathForum question that suggested that one could calculate the volume of the kinds of solids described above by taking the hyperarea of the base and multiplying it by the average of the heights, while technically on the mark, its wording sent me down the wrong road.

To find the volume of these kinds of solids one needs to multiply the volume formula for a regular simplex (also shown in the original post):

enter image description here

times the average of the heights of the cap.

Calculating a solid with 3 heights to the cap will need a 2 dimensional regular simplex as a base (an equilateral triangle). In this case the formula for the volume of a regular simplex will give the triangle's area, hence the confusion about hyperarea.

When calculating for heights of n dimensions, the base simplex will have n-1 dimensions.

So, a little Mathematica function will do the trick:

volSimplexWithHyperCap[heights_List] := Module[{n, simplexVolume},
  n = Length[heights] - 1;
  simplexVolume = Sqrt[n + 1]/(n! * Sqrt[2^(n)]);
  N[simplexVolume * Mean[heights]]]

volSimplexWithHyperCap[3, {1, 1, 1}]

0.433013

This corresponds correctly to the prism volume in the question. It also, readily extends to higher dimensions:

volSimplexWithHyperCap[{1, 3, 2, 3, 5}]
0.0652186

This has the advantage over integration of being relatively simple and very fast to calculate. Not a Cholesky decomposition solution like I imagined at first, but it looks serviceable.

Maybe @MvG will think of a proof.

Thanks to everyone for their patients with all the edits.

[Math] 3D Convex Hull and The Gift Wrapping Principle

Comment was too short.

Plane is uniquely determined by three non-collinear points. Imagine a door, it behaves like a plane determined by two points (the hinges), however, if you add yet another point (e.g. hold it by hand), then the door cannot move (unless you hold it exactly at the rotation axis). The edge from the quoted passage is like the door rotation axis (the hinges are the two vertices that the edge connects), now you need a third point (one of $C$ obviously), and one such that all others are on one side of the door. Naturally, the solution is to open the door as wide as you need (i.e. use the largest convex angle).

I hope this helps $\ddot\smile$

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Related Solutions

[Math] Volume of n-dimensional solid w/ n-1 dimensional simplex as a base

[Math] 3D Convex Hull and The Gift Wrapping Principle

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