Centroid

The centroid of a body is its geometric centre. We have a few key formulations of the centroid. For a finite set of $k$ points in ${\mathbb{R}}^n$, the centroid is given by:

$$\vec{C}=\frac{{\vec{x}}_1+{\vec{x}}_2+\cdots +{\vec{x}}_k}{k}$$

i.e., it is the arithmetic mean of each dimension.

In classical mechanics, we are often also concerned with the centre of mass and centre of gravity. When the body has a constant density or specific weight, the centre of gravity is the centroid of the body. We can calculate this with a composite method, where:

$$\bar{x}=\frac{\Sigma \tilde{x}W}{\Sigma W},\bar{y}=\frac{\Sigma \tilde{y}W}{\Sigma W},\bar{z}=\frac{\Sigma \tilde{z}W}{\Sigma W}$$

$⟨\bar{x},\bar{y},\bar{z}⟩$ are the coordinates of the centre of gravity of the body. $⟨\tilde{x},\tilde{y},\tilde{z}⟩$ are the coordinates of the centre of gravity of each composite part of the body. $\Sigma W$ is the sum of the weights of all composite parts of the body, i.e., its total weight. $\tilde{x}W$ is the contribution associated to a certain component’s weight.