# 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**.

## 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 - 1933: D.M.Y. Sommerville:
*Analytical Conics*(3rd ed.) ... (previous) ... (next): Chapter $\text I$. Coordinates: $12$. Mean points and centres of gravity