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