Definition:Centroid/Weighted Set of Points

Definition

Let $S = \set {A_1, A_2, \ldots, A_n}$ be a set of $n$ points in Euclidean space whose position vectors are given by $\mathbf a_1, \mathbf a_2, \dotsc, \mathbf a_n$ repectively.

Let $W: S \to \R$ be a weight function on $S$.

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

$\vec {OG} = \dfrac {w_1 \mathbf a_1 + w_2 \mathbf a_2 + \dotsb + w_n \mathbf a_n} {w_1 + w_2 + \dotsb + w_n}$

where $w_i = \map W {A_i}$ for each $i$.

Then $G$ is known as the centroid of $S$ with weights $w_i, w_2, \dotsc, w_n$.

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.

The centroid of $S$ with weights $w_i, w_2, \dotsc, w_n$ is a clumsy term, and a more streamlined one is to be sought.

In 1921: C.E. Weatherburn: Elementary Vector Analysis it is actually referred to as the centroid of the given points with associated numbers $w_1, w_2, \ldots, w_n$ respectively.

However, in the same work, the author does introduce the word strength in a footnote, which he suggests "may be found more convenient" than associated number.

Also see

• Results about centroids can be found here.