Definition:Centroid of Set of Points/Definition 2

Definition

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

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

• Equivalence of Definitions of Centroid of Set of Points

• Results about centroids can be found here.

Sources

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