TOPICS
Search

Polyhedron Centroid

The polyhedron centroid is the geometric centroid of a polyhedron. For a closed polyhedron with volume V and N triangular faces having vertices (a_i,b_i,c_i), it can be computed using the divergence theorem as

x^_=1/(48V)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/(48V)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/(48V)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)

The vertices of each face must be ordered so that n_i points outward. These formulas apply to polyhedra with arbitrary faces, since faces having more than three vertices can be triangulated. They apply to convex polyhedra as well as nonconvex polyhedra.

The centroid can also be computed using the divergence theorem by integrating the vector fields 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 and dividing the resulting surface integrals by V.

See also

Divergence Theorem, Geometric Centroid, Polyhedron, Polyhedron Volume

Explore with Wolfram|Alpha

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://nurnberg.maths.unitn.it/centroid.pdf.

Referenced on Wolfram|Alpha

Polyhedron Centroid

Cite this as:

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

Subject classifications