Definition:Centroid of Set of Points/Definition 1

Definition

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

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 $S$.

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.

Beware that some sources use the term center of gravity even when approaching the topic from a pure mathematical perspective, which can cause confusion.

Also see

• Equivalence of Definitions of Centroid of Set of Points

• Results about centroids can be found here.

Sources

• 1921: C.E. Weatherburn: Elementary Vector Analysis ... (previous) ... (next): Chapter $\text I$. Addition and Subtraction of Vectors. Centroids: Centroids: $9$. Centroid, or centre of mean position