# Definition:Centroid/Set of Points

## Definition

Let $S = \set {A_1, A_2, \ldots, A_n}$ be a set of $n$ points in Euclidean space.

### Definition 1

Let the position vectors of the elements of $S$ be given by $\mathbf a_1, \mathbf a_2, \dotsc, \mathbf a_n$ respectively.

Let $G$ be the point whose position vector is given by:

$\vec {OG} = \dfrac 1 n \paren {\mathbf a_1 + \mathbf a_2 + \dotsb + \mathbf a_n}$

Then $G$ is known as the centroid of $S$.

### Definition 2

Let the Cartesian coordinates of the elements of $S$ be $\tuple {x_j, y_j, z_j}$ for each $j \in \set {1, 2, \ldots, n}$.

Let $G$ be the point whose Cartesian coordinates are given by:

$G = \tuple {\dfrac 1 n \ds \sum_{j \mathop = 1}^n x_j, \dfrac 1 n \ds \sum_{j \mathop = 1}^n y_j, \dfrac 1 n \ds \sum_{j \mathop = 1}^n z_j}$

That is, the arithmetic mean of the Cartesian coordinates of the elements of $S$

Then $G$ is known as the centroid of $S$.

## Also known as

A centroid is also referred to as a center of mean position.

Some sources refer to it as a mean point.

Approaches to this subject from the direction of physics and mechanics can be seen referring to it as a center of gravity.

However, it needs to be noted that the latter is merely a special case of a centroid.

## Also see

• Results about centroids can be found here.