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Polyhedron Centroid

The geometric centroid of a polyhedron composed of N triangular faces with vertices (a_i,b_i,c_i) can be computed using the curl theorem as

x^_=1/(48)sum_(i=1)^(n)n_i·x^^([(a_i+b_i)·x^^]^2+[(b_i+c_i)·x^^]^2+[(c_i+a_i)·x^^]^2)
(1)
y^_=1/(48)sum_(i=1)^(n)n_i·y^^([(a_i+b_i)·y^^]^2+[(b_i+c_i)·y^^]^2+[(c_i+a_i)·y^^]^2)
(2)
z^_=1/(48)sum_(i=1)^(n)n_i·z^^([(a_i+b_i)·z^^]^2+[(b_i+c_i)·z^^]^2+[(c_i+a_i)·z^^]^2),
(3)

where the normal n_i is given by the cross product

n_i=(b_i-a_i)x(c_i-a_i).
(4)

This formula can be applied to polyhedra with arbitrary faces since faces having more than three vertices can be triangulated. Furthermore, the formula applies to concave polyhedra as well as convex ones.

The centroid can also be computed using the divergence theorem by integrating the functions F=(x^2/2,0,0), G=(xy,0,0), and H=(xz,0,0) which have divergence del ·F=x, del ·G=y, del ·H=z everywhere, over the triangulated faces of the polyhedron.

See also

Geometric Centroid, Polyhedron, Polyhedron Volume,

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References

Dobrovolskis, A. R. "Inertia of Any Polyhedron." Icarus 124, 698-704, 1996.Lawlor, O. "Boundary Integration and the Rotational Inertia Matrix." CS 482 Lecture. https://www.cs.uaf.edu/2015/spring/cs482/lecture/02_20_boundary.html.Mirtich, B. "Fast and Accurate Computation of Polyhedral Mass Properties." J. Graphics Tools 1, No. 2, 31-50, Feb. 1996.Nürnberg, R. "Calculating the Area and Centroid of a Polygon in 2D." 2013. https://www.ma.imperial.ac.uk/~rn/centroid.pdf.

Cite this as:

Weisstein, Eric W. "Polyhedron Centroid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PolyhedronCentroid.html

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