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