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 $X$.
The geometric centroid of a physical object coincides with its 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. These conditions are sufficient but not necessary.
The centroid of a finite set of points can be computed as the arithmetic mean of each coordinate of the points.
In geography, the centroid of a region of the Earth's surface is often known as its 'geographic center'.

Contents

Centroid of triangle and tetrahedron

Centroids of cones and pyramids

Centroid and convexity

Integral formula

Center of symmetry

See also

External links

Centroid of triangle and tetrahedron

Triangle centroid 1.svg Triangle centroid 2.svg

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, which is to say it is located ⅓ of the perpendicular distance between each side and the opposing point. (As illustrated in the figures to the right).
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. That is, if the three vertices are located at $(x\_a,\; y\_a)$, $(x\_b,\; y\_b)$, and $(x\_c,\; y\_c)$, then the centroid is at
:$Big($
egin{matrix}rac13end{matrix} (x_a+x_b+x_c),;
egin{matrix}rac13end{matrix} (y_a+y_b+y_c) Big)
= egin{matrix}rac13end{matrix} (x_a, y_a)
+ egin{matrix}rac13end{matrix} (x_b, y_b)
+ egin{matrix}rac13end{matrix} (x_c, y_c).
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. If the set of vertices of a simplex is $\{v\_0,...,v\_n\}$, then considering the vertices as vectors, the centroid is at
:
The isogonal conjugate of a triangle's centroid is its symmedian point.

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.
A centroid is simply where the 3 medians of a triangle intersect.

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.

Integral formula

The abscissa (x-coordinate) 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 $x$. This formula can be derived from the first moment about the y-axis of the area.
This process is equivalent to taking a weighted average. Supposing that the y-axis represents frequency, and the x-axis represents the variable whose average we want to find, then the location of the centroid along the x-axis is simply the mean:
Hence the centroid can be thought of as a weighted average of many infintesimally small elements that represent a particular shape.
The same formula yields the first coordinate of the centroid of an object in $R^n$, 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. In the special case where f is normalized, i. e. the denominator is 1, the centroid is called the mean of f.
The formula cannot be applied if the object has zero measure, or if either integral diverges.

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.

See also

★ List of centroids

★ Pappus's centroid theorem

External links

★ ''Encyclopedia of Triangles Centers'' by Clark Kimberling. The centroid is indexed as X(2).

★ Triangle centers by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.

★ Characteristic Property of Centroid at cut-the-knot

★ Barycentric Coordinates at cut-the-knot

★ Centroid of a triangle With interactive animation