PlanetMath: betweenness in rays

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betweenness in rays

(Definition)

Let be a linear ordered geometry. Fix a point and consider the pencil $\Pi(p)$ of all rays emanating from it. Let $\alpha\neq\beta \in\Pi(p)$ . A point is said to be an interior point of $\alpha$ and $\beta$ if there are points $s\in\alpha$ and $t\in\beta$ such that

and are on the same side of line $\overleftrightarrow{pt}$ , and
and are on the same side of line $\overleftrightarrow{ps}$ .

A point

is said to be between $\alpha$ and $\beta$ if there are points $s\in\alpha$ and $t\in\beta$ such that

is between

and

. A point that is between two rays is an interior point of these rays, but not vice versa in general. A ray $\rho\in\Pi(p)$ is said to be between rays $\alpha$ and $\beta$ if there is an interior point of $\alpha$ and $\beta$ lying on $\rho$ .

Properties

Suppose $\alpha,\beta,\rho\in\Pi(p)$ and $\rho$ is between $\alpha$ and $\beta$ . Then
1. any point on $\rho$ is an interior point of $\alpha$ and $\beta$ ;
2. any point on the line containing $\rho$ that is an interior point of $\alpha$ and $\beta$ must be a point on $\rho$ ;
3. there is a point on $\rho$ that is between $\alpha$ and $\beta$ . This is known as the Crossbar Theorem, the two “crossbars” being $\rho$ and a line segment joining a point on $\alpha$ and a point on $\beta$ ;
4. if is defined as above, then any point between and is between $\alpha$ and $\beta$ .
There are no rays between two opposite rays.
If $\rho$ is between $\alpha$ and $\beta$ , then $\beta$ is not between $\alpha$ and $\rho$ .
If $\alpha,\beta\in\Pi(p)$ has a ray $\rho$ between them, then $\alpha$ and $\beta$ must lie on the same half plane of some line.
The converse of the above statement is true too: if $\alpha,\beta\in\Pi(p)$ are distinct rays that are not opposite of one another, then there exist a ray $\rho\in\Pi(p)$ such that $\rho$ is between $\alpha$ and $\beta$ .
Given $\alpha,\beta\in\Pi(p)$ with $\alpha\neq\beta$ and $\alpha\neq-\beta$ . We can write $\Pi(p)$ as a disjoint union of two subsets:
1. $A =\lbrace \rho\in\Pi(p)\mid \rho$ is between $\alpha$ and $\beta\rbrace$ ,
2. $B=\Pi(p)-A$ .
Let $\rho,\sigma\in\Pi(p)$ be two rays distinct from both $\alpha$ and $\beta$ . Suppose $x\in\rho$ and $y\in\sigma$ . Then $\rho,\sigma$ belong to the same subset if and only if $\overline{xy}$ does not intersect either $\alpha$ or $\beta$ .

Bibliography

1: D. Hilbert, Foundations of Geometry, Open Court Publishing Co. (1971)
2: K. Borsuk and W. Szmielew, Foundations of Geometry, North-Holland Publishing Co. Amsterdam (1960)
3: M. J. Greenberg, Euclidean and Non-Euclidean Geometries, Development and History, W. H. Freeman and Company, San Francisco (1974)

"betweenness in rays" is owned by CWoo. [ full author list (2) ]

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See Also: angle, ray

Also defines:

interior point, between rays, between two rays, crossbar theorem

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Cross-references: intersect, subsets, disjoint union, opposite, converse, half plane, lie on, opposite rays, line segment, line, properties, lying on, side of line, rays, pencil, point, fix, linear ordered geometry
There are 6 references to this entry.

This is version 3 of betweenness in rays, born on 2005-10-27, modified 2007-06-22.
Object id is 7449, canonical name is BetweennessInRays.
Accessed 4111 times total.

Classification:

AMS MSC:	51G05 (Geometry :: Ordered geometries )
	51F20 (Geometry :: Metric geometry :: Congruence and orthogonality)

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