Finding the moment of inertia of a 2D polygon.

[Droctagonopus]

Hi everyone,

Is there a general method for finding out the moment of inertia of an irregular convex 2D polygon given the coordinates of its vertices?

I have thought of one possible method:
Split the polygon into multiple triangles and find the moment of inertia of each triangle around the centre of mass of the original polygon using the parallel axis theorem. (Is the formula for moment of inertia of a triangle bh^3/12? Is there a more vertices-friendly way of finding the moment of inertia?)

Is there a better method to do it? I'm trying to make this computer algorithm friendly so a method with repetition of similar calculations would be helpful.

 

[Simon Bridge]

Nope - that's pretty much it.
http://lab.polygonal.de/?p=57

 

[SteamKing]

There's a method which can be derived from Green's Theorem in the plane by which a closed curve described by piecewise line segments can have its area and first and second moments of area calculated by using only the (x,y) coordinates of the endpoints of the line segments as input.

See this article, p. 8, for the formulas: http://www.infogoaround.org/JBook/CentroidInertia.pdf

The curve is described starting at an arbitrary endpoint of one of the line segments and proceeding counterclockwise around the area of interest until reaching the starting point. Counterclockwise orientation produces positive areas and moments while clockwise orientation produces negative areas and moments, so that polygons with holes can also be handled by this method.

The areas and moments are referenced about the coordinate axes, so the parallel axis theorem is required to determine centroidal values of these quantities once you have completed the circuit of the polygon.

Calculation using individual triangles is not necessary.

See also:
http://en.wikipedia.org/wiki/Polygon
http://www.seas.upenn.edu/~sys502/extra_materials/Polygon%20Area%20and%20Centroid.pdf