Centroid

Definition

Let $k$ be a positive integer, and let $K$ be a field in which $k \neq 0$ . Let $A$ be an affine space over a $K$ -vector space, and let $p_1, \dots, p_k$ be points of $A$ . The centroid of $p_1, \dots, p_k$ is the affine combination with all weights equal to $\frac{1}{k}$ : $G = \frac{1}{k} p_1 + \frac{1}{k} p_2 + \dots + \frac{1}{k} p_k.$ The centroid of a triangle $ABC$ is the centroid $G = \frac{1}{3}(A + B + C)$ of its vertices.

Remarks

In $\mathbb{R}^n$ , the coordinates of the centroid are the means of the coordinates of the points. In $\mathbb{R}^n$ , the centroid is the point $x$ that makes the sum $\sum_i \|x - p_i\|^2$ smallest. The centroid of two points is their midpoint. The centroid of a triangle is also the centre of mass of a uniform triangular plate.

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Affine combination concept
Centroid definition

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Monge point

Tag	`0Y5RHNRKJZ`
URI	`m2://0Y5RHNRKJZ`
Kind	definition
Writing	W1 stated
MSC	51M04
Defines	Centroid
Uses	Affine combination

Centroid

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