Midpoint

Definition

Let $K$ be a field in which $2 \neq 0$ , and let $A$ be an affine space over a vector space $V$ over $K$ . Let $p$ and $q$ be points of $A$ . The midpoint of $p$ and $q$ is the point $\operatorname{midpoint}(p, q) = p + \tfrac{1}{2}(q - p).$ It is the affine combination $\frac{1}{2} p + \frac{1}{2} q$ .

Remarks

In $\mathbb{R}^n$ , the midpoint of $(p_1, \dots, p_n)$ and $(q_1, \dots, q_n)$ is $\left(\frac{p_1 + q_1}{2}, \dots, \frac{p_n + q_n}{2}\right)$ . In a Euclidean space, the midpoint $m$ of $p$ and $q$ satisfies $d(p, m) = d(m, q) = \frac{1}{2} d(p, q)$ . It is the only point with this property. The midpoint is symmetric: $\operatorname{midpoint}(p, q) = \operatorname{midpoint}(q, p)$ . Also $\operatorname{midpoint}(p, p) = p$ .

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Affine combination concept
Midpoint definition

Tag	`06F3Y6CMG9`
URI	`m2://06F3Y6CMG9`
Kind	definition
Writing	W1 stated
MSC	51N10
Defines	Midpoint
Uses	Affine combination

Midpoint

Definition

Remarks

Reading path

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