ray

ray

Rays on the

real line

. A

ray

on the real line

ℝ

is just an

open set

of the form

(

p

,

∞

)

, or

(

-

∞

,

p

)

. A ray is also called a

half line

, or an

open ray

, to distinguish the notion of a

closed ray

, which includes its

endpoint

.

Properties

Suppose

p

,

q

∈

ℝ

and

p

≤

q

.

•

(

p

,

∞

)

∩

(

q

,

∞

)

=

(

q

,

∞

)

.

•

(

p

,

∞

)

∪

(

q

,

∞

)

=

(

p

,

∞

)

.

•

(

p

,

∞

)

∩

(

-

∞

,

q

)

=

(

p

,

q

)

if

p

≠

q

, and

∅

if

p

=

q

.

•

(

-

∞

,

p

)

∩

(

q

,

∞

)

=

∅

.

•

(

p

,

∞

)

∪

(

-

∞

,

q

)

=

ℝ

if

p

≠

q

, and

ℝ

-

{

p

}

if

p

=

q

.

•

(

-

∞

,

p

)

∪

(

q

,

∞

)

=

ℝ

-

[

p

,

q

]

if

p

≠

q

, and

ℝ

-

{

p

}

if

p

=

q

.

Rays in a general

Euclidean space

. Let

ℓ

be a line in

ℝ

n

and let

p

be a point

lying on

the

ℓ

. We may parameterize

ℓ

=

ℓ

⁢

(

t

)

(parameter

t

∈

ℝ

) so that

ℓ

⁢

(

0

)

=

p

. An (

open

)

ray

ρ

lying on

ℓ

with

endpoint

p

is the set of points

ρ

=

{

r

⁢

∣

r

=

ℓ

⁢

(

t

)

,

t

>

⁢

0

}

.

If the

inequality

t

>

0

is relaxed to

t

≥

0

in the above expression, then we have a

closed ray

. Note that if the inequality above were changed to

t

<

0

instead, we end up again with a ray lying on

ℓ

and endpoint

p

. It is a ray because we can reparameterize

ℓ

by using the parameter

s

=

-

t

instead, so that

{

r

∣

r

=

ℓ

⁢

(

t

)

,

t

<

0

}

=

{

r

⁢

∣

r

=

ℓ

⁢

(

s

)

,

s

>

⁢

0

}

.

The

difference

between the two rays is that they point in the opposite directions. Therefore, in general, a ray can be
characterized by

•

a line,

•

a point lying on the line, and

•

a direction on the line.

Rays in an

ordered geometry

: Given two distinct points

p

,

q

in an ordered geometry

(

A

,

B

)

(

A

is the underlying incidence geometry (

http://planetmath.org/IncidenceGeometry

) and

B

is the

strict betweenness relation

defined on the points of

A

). The set

p

⁢

q

¯

∪

{

q

}

∪

{

r

∣

q

∈

p

⁢

r

¯

}

,

where

s

⁢

t

¯

denotes the

open line segment

with endpoints

s

and

t

, is called the

(open) ray generated by

p

and

q

emanating from

p

. It is denoted by

p

⁢

q

→

.

p

in

p

⁢

q

→

is called the

source

or the

end point

of the ray. A

closed ray generated by

p

and

q

with endpoint

p

is the set

p

⁢

q

→

∪

{

p

}

.

Properties.

•

for any point

r

∈

p

⁢

q

→

,

p

⁢

r

→

=

p

⁢

q

→

.

•

p

⁢

q

→

∪

q

⁢

p

→

=

p

⁢

q

↔

and

p

⁢

q

→

∩

q

⁢

p

→

=

p

⁢

q

¯

. We say that a ray lies on a line if all of the points in the ray are

incident

with the line. Also, a

line segment

lies on a ray if it is a subset of the ray.

•

The

opposite ray

of

ρ

=

p

⁢

q

→

is defined to be

q

⁢

p

→

-

p

⁢

q

¯

-

{

q

}

.

It is denoted by

-

ρ

.

•

The opposite ray

-

ρ

of a ray

ρ

is ray. Suppose

ρ

=

p

⁢

q

→

. Then

ρ

has the property that

(a)

ρ

∩

(

-

ρ

)

=

∅

and

(b)

ρ

∪

(

-

ρ

)

=

p

⁢

q

↔

-

{

p

}

.

Conversely

, given a ray

ρ

=

p

⁢

q

→

, any ray

ρ

′

satisfying the above two properties (replacing

-

ρ

by

ρ

′

) is the opposite ray of

ρ

.

•

Given any point

p

on a line

ℓ

, there are exactly two rays lying on

ℓ

with endpoint

p

. Furthermore,

p

is between

q

and

r

in

ℓ

iff

q

and

r

lie on opposite rays on

ℓ

.

•

Given any two rays

ρ

and

ϱ

, exactly one of the following holds:

(a)

ρ

∩

ϱ

=

∅

,

(b)

ρ

∩

ϱ

=

a line segment, or

(c)

ρ

∩

ϱ

=

a ray.

It is not hard to see that in the last case, one ray is included in the other, and their

intersection

is the “smaller” of the two rays. In the first two cases, the two rays are said to be (pointing) in the opposite direction. In the last case, the two are said to be in the same direction. Opposite rays are clearly pointing in the opposite direction.

•

An

equivalence relation

can be defined on the set of all rays lying on a line

ℓ

by whether they are pointing in the same direction or not. Thus, the set of all rays lying on

ℓ

can be partitioned into two subsets

R

and

R

′

, so that if

ρ

,

ϱ

∈

R

(or

R

′

), then they are pointing in the same direction; and if

ρ

∈

R

and

ϱ

∈

R

′

are pointing in the opposite direction.

•

Pick one of the two subsets from above, say

R

. Define

≤

on

R

by

ρ

≤

ϱ

if

ρ

⊆

ϱ

. Then

≤

is a

linear order

on

R

. This

≤

induces a linear order

≤

ℓ

on the line

ℓ

in the following way:

p

≤

ℓ

q

if the corresponding rays

ρ

,

ϱ

∈

R

, with endpoints

p

and

q

respectively, we have

ρ

≤

ϱ

. This is one way to define a

linear ordering

on a line

ℓ

. An alternative, but

equivalent

way of defining a linear ordering on a line in an ordered geometry can be found in the entry under ordered geometry.

•

Note that in defining

≤

, we could have used

R

′

instead of

R

. This is an example of the

duality

of linear ordering.

References

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)

Title

ray

Canonical name

Ray

Date of creation

2013-03-22 15:28:36

Last modified on

2013-03-22 15:28:36

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51G05

Synonym

half line

Synonym

open ray

Related topic

BetweennessInRays

Defines

closed ray

Defines

opposite ray

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