Centroid: Encyclopedia - Centroid

Centroid

In geometry, the centroid or barycenter of an object X in n-dimensional space is the intersection of all hyperplanes that divide X into two parts of equal moment about the hyperplane. Informally, it is the "average" of all points of X.

In physics, the words centroid and barycenter may mean either the center of mass or the center of gravity of an object, depending on context. The centroid may also be the geometric center, as above; which may or may not coincide with the other two points.

Centroid - Centroid of triangle and tetrahedon

The centroid of a triangle is the point of intersection of its medians (the lines joining each vertex with the midpoint of the opposite side). The centroid divides each of the medians in the ratio 2:1.

The centroid is the triangle's center of mass if the triangle is made from a uniform sheet of material. Its Cartesian coordinates are the means of the coordinates of the three vertices.

A similar result holds for a tetrahedron: its centroid is the intersection of all line segments that connect each vertex to the centroid of the opposite face. These line segments are divided by the centroid in the ratio 3:1. The result generalizes to any n-dimensional simplex in the obvious way.

The isogonal conjugate of a triangle's centroid is its symmedian point.

Centroid - Centroids of cones and pyramids

The centroid of a cone or pyramid is located on the line segment that connects the apex to the centroid of the base, and divides that segment in the ratio 3:1.

Centroid - Centroid and convexity

The centroid of a convex object always lies in the object. A concave object might have a centroid that is outside the figure itself. The centroid of a ring or a bowl, for example, lies in the object's central void.

Centroid - Integral formula

The abscissa of the centroid of a plane figure can be given as the integral , where f(x) is the vertical extent of the object at abscissa x.

The same formula yields the first coordinate of the centroid of an object in , for any dimension n, provided that f(x) is the (n − 1)-dimensional measure of the object's cross-section at coordinate x — that is, the set of all points in the object whose first coordinate is x.

Note that the denominator is simply the object's n-dimensional measure. The formula cannot be applied if the object has zero measure, or if either integral diverges.

Centroid - Center of symmetry

If the centroid is defined, it is a fixed point of all isometries in its symmetry group. Thus symmetry may fully or partially determine the centroid, depending on the kind of symmetry. It also follows that for an object with translational symmetry the centroid is undefined, because a translation has no fixed point.

Centroid - Physical centroids

The geometric centroid of a physical object coincides with the center of mass if the object has uniform density, or if the object's shape and density have a symmetry which fully determines the centroid. In turn, the center of mass coincides with the center of gravity if the object is in a uniform gravitational field. These conditions are sufficient but not necessary.

See also

• Pappus's centroid theorem

Adapted from the Wikipedia article "Centroid", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki