Definitions

# Centroid

[sen-troid]
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 centroid of an 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 known as its geographical center.

The centroid of a convex object always lies in the object. A non-convex 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 of triangle and tetrahedron

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 $\left(x_a, y_a\right)$, $\left(x_b, y_b\right)$, and $\left(x_c, y_c\right)$, then the centroid is at:

$Big\left($
begin{matrix}frac13end{matrix} (x_a+x_b+x_c),; begin{matrix}frac13end{matrix} (y_a+y_b+y_c) Big) = begin{matrix}frac13end{matrix} (x_a, y_a) + begin{matrix}frac13end{matrix} (x_b, y_b) + begin{matrix}frac13end{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 $\left\{v_0,...,v_n\right\}$, then considering the vertices as vectors, the centroid is at:

$frac\left\{1\right\}\left\{n+1\right\}sum_\left\{i=0\right\}^n v_i$

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

### Proof that the centroid of a triangle divides each median in the ratio 2:1

Let the medians AD, BE and CF of the triangle ABC intersect at G, the centroid of the triangle, and let the straight line AD be extended up to the point O such that

$AG = GO. ,$

Then the triangles AGE and AOC are similar (common angle at A, AO is twice AG, AC is twice AE), and so OC is parallel to GE. But GE is BG extended, and so OC is parallel to BG. Similarly, OB is parallel to CG.

The figure GBOC is therefore a parallelogram. Since the diagonals of a parallelogram bisect one another, the point of intersection D between the diagonals GO and BC is such that GD = DO, and

$GO = GD + DO = 2GD. ,$

So, $AG = GO = 2GD, ,$

or $AG:GD = 2:1. ,$

This is true for every other median.

## Centroid of polygon

The centroid of a non-overlapping closed polygon defined by N vertices (xi , yi ) can be calculated as follows. The notional vertex (xN , yN ) is the same as (x0 , y0 ).

The area of the polygon is given by:

$A = frac\left\{1\right\}\left\{2\right\}sum_\left\{i=0\right\}^\left\{N-1\right\} \left(x_i y_\left\{i+1\right\} - x_\left\{i+1\right\} y_i\right)$

The centroid of the polygon is then given by:

$C_x = frac\left\{1\right\}\left\{6A\right\}sum_\left\{i=0\right\}^\left\{N-1\right\}\left(x_i+x_\left\{i+1\right\}\right)\left(x_i y_\left\{i+1\right\} - x_\left\{i+1\right\} y_i\right)$

$C_y = frac\left\{1\right\}\left\{6A\right\}sum_\left\{i=0\right\}^\left\{N-1\right\}\left(y_i+y_\left\{i+1\right\}\right)\left(x_i y_\left\{i+1\right\} - x_\left\{i+1\right\} y_i\right)$

## Centroid of a finite set of points

Given a finite set of points $x_1,x_2,ldots,x_k$ in $mathbb\left\{R\right\}^n$, their centroid $C$ is defined to be

$C = frac\left\{x_1+x_2+cdots+x_k\right\}\left\{k\right\}$.

## Area centroid

The centroid of an area is very similar to the center of mass of a body. This is calculated using only the geometry of the figure. If the body is homogeneous, the center of mass will be at the centroid.

For a two body figure, you may have an equation that looks like this:

$overline\left\{y\right\} = dfrac\left\{overline\left\{y_1\right\}A_1 + overline\left\{y_2\right\}A_2\right\}\left\{A_1 + A_2\right\}$

$overline\left\{y\right\}$ is the distance from your reference coordinate axis to the centroid of the particular area. $A$ is the area of that particular section.

The general function for calculating the centroid of a geometrically complex cross section is most easily applied when the figure is divided into known simple geometries and then applying the formula:

$overline\left\{x\right\} = frac\left\{sum overline\left\{x_i\right\}A_i\right\}\left\{sum A_i\right\}$

$overline\left\{y\right\} = frac\left\{sum overline\left\{y_i\right\}A_i\right\}\left\{sum A_i\right\}$

The distance from the y-axis to the centroid is $overline\left\{x\right\}$. The distance from the x-axis to the centroid is $overline\left\{y\right\}$. The coordinates of the centroid are $\left(overline\left\{x\right\} , overline\left\{y\right\}\right)$.

## Integral formula

The abscissa (x coordinate) of the centroid of a plane figure can be given as the integral

$C_x = frac\left\{int x f\left(x\right) ; dx\right\}\left\{int f\left(x\right) ; dx\right\},$

where f(x) is the extent of the object along the y axis at abscissa x, that is the measure of the figure's section at 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: $bar\left\{x\right\}$

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.

## Centroid of cone and pyramid

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.

## 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.

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