Betweenness

Definition

Let $F$ be an ordered field, let $V$ be a vector space over $F$ , and let $P$ be an affine space over $V$ . Let $x, y, z \in P$ . The point $y$ is weakly between $x$ and $z$ if $y = x + t(z - x)$ for some $t \in F$ with $0 \le t \le 1$ . The point $y$ is strictly between $x$ and $z$ if it is weakly between them and $y \ne x$ and $y \ne z$ .

Remarks

Example: in $\R^2$ , the point $(1, 1)$ is strictly between $(0, 0)$ and $(2, 2)$ , with $t = 1/2$ . Non-example: $(3, 3)$ is on the line through $(0, 0)$ and $(2, 2)$ , but it is not between them, since it needs $t = 3/2 > 1$ . Weak betweenness is symmetric: if $y$ is weakly between $x$ and $z$ , then it is weakly between $z$ and $x$ , with $1 - t$ in place of $t$ . The set of points weakly between $x$ and $z$ is the closed segment $[x, z]$ . In axiomatic geometry, betweenness is not defined but taken as a primitive relation that satisfies the axioms of order.

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Affine space concept
Ordered field concept
Betweenness definition

Tag	`9CCSBY0JQW`
URI	`m2://9CCSBY0JQW`
Kind	definition
Writing	W0 stub
MSC	51G05, 51N10
Defines	Betweenness
Uses	Affine space, Ordered field

Betweenness

Definition

Remarks

Reading path

Further reading

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