ordered geometry

ordered geometry

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line segment

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generated by

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opposite sides

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end points

1

Definition

Let

(

A

,

B

)

be a

linear ordered geometry

, where

A

=

(

P

,

n

,

I

)

is an

incidence geometry

,
and

B

is a

strict betweenness relation

.
Recall that

P

is partitioned into

disjoint sets

P

0

,

…

,

P

n

, where

n

is a

positive integer

.

For

0

≤

i

<

n

, let

B

i

=

{

(

p

,

a

,

q

)

∈

P

0

×

P

i

×

P

0

∣

p

,

q

do not lie on

a

,

and there
exists a point

r

lying on

a

such that

(

p

,

r

,

q

)

∈

B

}

,

and

B

i

⁢

(

a

)

=

{

(

p

,

q

)

∣

(

p

,

a

,

q

)

∈

B

i

}

.

For any

a

∈

P

i

, the set is

symmetric

and anti-reflexive.

We say that the

hyperplane

a

∈

P

i

is
between

p

and

q

if

(

p

,

q

)

∈

B

i

⁢

(

a

)

.

We see that

B

0

=

B

.

Let’s look at the case when

i

=

1

. If

(

p

,

ℓ

,

q

)

∈

B

1

where

ℓ

is a line, then

p

,

q

and

ℓ

necessarily lie on a common
plane

π

.

The above diagram seems to suggest that

ℓ

“separates

π

into two regions”. However, this is not true in general without the next axiom.

An

ordered geometry

(

A

,

B

)

is a linear ordered geometry such that

S1

for any three non-collinear points

p

,

q

,

r

, and any line

ℓ

lying on the same plane

π

generated by

p

,

q

,

r

, if

(

p

,

q

)

∈

B

1

⁢

(

ℓ

)

and if

r

does not lie on

ℓ

, then at least one of

(

q

,

r

)

,

(

r

,

p

)

∈

B

1

⁢

(

ℓ

)

.

In fact, in axiom S1, it can be shown that exactly one of

(

q

,

r

)

and

(

r

,

p

)

is in

B

1

⁢

(

ℓ

)

. This axiom says that “a line lying on a plane separates the plane into two mutually exclusive subsets”.

Each subset is called an (open)

half plane

of the line.

A closed half plane is just the union of one of its open half planes and the line itself.

Suppose points

p

,

q

and line

ℓ

lie on plane

π

and that

ℓ

is between

p

and

q

. Then we say that

p

and

q

are on the

opposite sides of line

ℓ

. Two points are on the

same

side of line

ℓ

if they are not on the opposite sides of

ℓ

. If

r

is a third point (distinct from

p

,

q

) that lies on

π

and not on

ℓ

, then according to axiom S1 above,

r

must be on the same side of either

p

or

q

(but not both!). Same sidedness is an

equivalence relation

on points of

A

.

3
An

equivalent

characterization

of axiom S1 is in the form of Pasch’s

theorem

.

The ten conditions or axioms (seven betweenness, two

collinearity

,
and one “

separation

” axioms) are sometimes called the “order axioms”
of

A

.

It is customary, in an ordered geometry, to identify each element of

P

by its shadow (

http://planetmath.org/IncidenceGeometry

) (a subset of

P

0

), and we shall do so in this discussion. A line, for example, will then consist of points that are

incident

with it, as opposed to an abstract element of

P

1

.
Hence, we shall also confuse the notation

p

⁢

q

↔

with

I

0

⁢

(

p

⁢

q

↔

)

.

2

Remarks

•

Law of Trichotomy

on a strict betweenness relation: Let

B

be a strict betweenness relation. If

p

,

q

,

r

are

collinear

, then
exactly one of

(

p

,

q

,

r

)

,

(

q

,

r

,

p

)

, or

(

r

,

p

,

q

)

∈

B

.

•

In an ordered geometry, one can define familiar

concepts

, such as a
line segment, a ray, even an angle, using the order axioms above. For example,

B

p

*

q

is called the

open line segment

between

p

and

q

, and is more commonly denoted by

p

⁢

q

¯

, or

(

p

,

q

)

. A

closed line segment

between

p

and

q

is just

{

p

}

∪

p

⁢

q

¯

∪

{

q

}

, denoted by

[

p

,

q

]

. From the third remark under

betweenness relation

,

p

⁢

q

¯

=

B

p

*

q

=

B

q

*

p

=

q

⁢

p

¯

. The points

p

and

q

are called the

end points

of

p

⁢

q

¯

.

•

A

ray

is defined to be

B

p

⁢

q

. For a more detailed discussion, see the entry on ray (

http://planetmath.org/Ray

).

•

p

⁢

q

↔

=

B

⁢

(

p

,

q

)

.

•

p

⁢

q

¯

⊂

p

⁢

q

↔

. The

inclusion

is strict,
since there exists a point

r

such that

(

p

,

q

,

r

)

∈

B

by order
axiom S1.

r

lies on the

p

⁢

q

↔

and is clearly distinct from
both

p

and

q

.

•

Any line segment

p

⁢

q

¯

in an ordered geometry, in to being orderable, is linearly orderable, thanks to the Law of Trichotomy.

•

It fact,

≤

, defined on a line segment, can be extended to a

linear order

defined on the line that includes the segment (see the last remark above on betweenness relation). This shows that every line in an ordered geometry can be

linearly ordered

.

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

R. Hartshorne,

Geometry

: Euclid and Beyond

, Springer (2000)

Title

ordered geometry

Canonical name

OrderedGeometry

Date of creation

2013-03-22 15:28:21

Last modified on

2013-03-22 15:28:21

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

43

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51G05

Synonym

open interval

Synonym

closed interval

Synonym

interval

Related topic

PaschsTheorem

Defines

half plane

Defines

side of line

Defines

open line segment

Defines

closed line segment

Defines

opposite sides

Defines

open half plane

Defines

closed half plane

Defines

end points

Defines

open line segment

Defines

closed line segment

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