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.
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
- Results about centroids can be found here.