# Definition:Centroid

## Definition

### Centroid of Set of Points

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 $S$.

### Centroid of Weighted Set of Points

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 $w_i, w_2, \dotsc, w_n$**.

### Centroid of Surface

Let $S$ be a surface.

Let $S$ be divided into a large number $n$ of small elements.

Consider one point of each of these elements.

Let a weight function be associated with this set of points.

Let $G$ be the centroid of each of these weighted points.

Let $n$ increase indefinitely, such that each element of $S$ converges to a point.

Then the limiting position of $G$ is the **centroid** of $S$.

### Centroid of Solid Figure

Let $F$ be a solid figure.

Let $F$ be divided into a large number $n$ of small elements.

Consider one point of each of these elements.

Let a weight function be associated with this set of points.

Let $G$ be the centroid of each of these weighted points.

Let $n$ increase indefinitely, such that each element of $F$ converges to a point.

Then the limiting position of $G$ is the **centroid** of $F$.

### Centroid of Triangle

Let $\triangle ABC$ be a triangle.

The **centroid** of $\triangle ABC$ is the point $G$ where its three medians $AL$, $MB$ and $CN$ meet.

## 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**.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**centroid**

This page may be the result of a refactoring operation.As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.In particular: The definition as actually given here is neededIf you have access to any of these works, then you are invited to review this list, and make any necessary corrections.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{SourceReview}}` from the code. |

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**centroid** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**centroid**