affine geometry

affine geometry

Affine Spaces

Let

V

be a left (right)

vector space

over a division ring

D

. An

affine space

associated with

V

is a set

A

, together with a

function

f

:

A

×

A

→

V

, with the following conditions:

1.

for every

P

∈

A

, the function

f

⁢

(

P

,

-

)

:

A

→

V

is onto.

2.

f

is

non-degenerate

in the sense that

f

⁢

(

P

,

Q

)

=

0

implies

P

=

Q

.

3.

f

⁢

(

P

,

Q

)

+

f

⁢

(

Q

,

R

)

=

f

⁢

(

P

,

R

)

.

An

affine space

is an affine space associated with some vector space. Elements of

A

are called

points

of

A

. The

dimension

of an affine space is just the dimension of its associated vector space. The function

f

is called a

direction

. We sometimes write

P

⁢

Q

⟶

for

f

⁢

(

P

,

Q

)

.

For example,

V

itself is an affine space associated with

V

, with

f

given by

f

⁢

(

v

,

w

)

=

w

-

v

.

Below are some properties of

f

:

1.

f

⁢

(

P

,

P

)

=

0

, because

f

⁢

(

P

,

P

)

+

f

⁢

(

P

,

P

)

=

f

⁢

(

P

,

P

)

by condition 3, hence the result.

2.

f

⁢

(

P

,

-

)

is

one-to-one

, and therefore a

bijection

, for if

f

⁢

(

P

,

Q

)

=

f

⁢

(

P

,

R

)

, then

f

⁢

(

P

,

Q

)

+

f

⁢

(

Q

,

R

)

=

f

⁢

(

P

,

R

)

=

f

⁢

(

P

,

Q

)

, so

f

⁢

(

Q

,

R

)

=

0

, or

Q

=

R

, by condition 2.

3.

f

⁢

(

P

,

Q

)

=

-

f

⁢

(

Q

,

P

)

, because

f

⁢

(

P

,

Q

)

+

f

⁢

(

Q

,

P

)

=

f

⁢

(

P

,

P

)

=

0

.

4.

f

⁢

(

-

,

Q

)

is also a bijection as a result.

Vector Spaces Induced by an Affine Space

An affine space is often called a “vector space without the origin”. In other words, singling out a point in an affine space gives us a vector space, in the following sense: fix a point

P

in an affine space

A

, define a vector space

A

P

as follows:

1.

vectors of

A

P

are points of

A

,

2.

define

+

P

:

A

P

×

A

P

→

A

P

by

Q

+

P

R

:=

S

, where

S

is the point determined by

f

⁢

(

P

,

Q

)

+

f

⁢

(

P

,

R

)

=

f

⁢

(

P

,

S

)

. Because of property 2 above,

S

is uniquely determined.

3.

define

⋅

P

:

D

×

A

P

→

A

P

by

d

⋅

P

Q

:=

T

, where

T

is the point determined by

d

⁢

f

⁢

(

P

,

Q

)

=

f

⁢

(

P

,

T

)

. Again,

T

is unique.

Both

+

P

and

⋅

P

are well-defined, because

f

⁢

(

P

,

-

)

is a bijection. When there is no confusion, we may drop the

subscript

P

. It is also easy to verify that

A

P

, together with

+

and

⋅

, is indeed a left vector space over

D

, with

P

as the origin, written

0

P

. Furthermore,

A

P

is

isomorphic

to

V

. Hence,

A

P

≅

A

Q

for any two points

P

,

Q

∈

A

; there is nothing special about

P

, and any point of

A

can be used as the origin of a vector space.

Affine Subspaces

Continue to assume that

V

is a left vector space over a division ring

D

, and

(

A

,

f

)

an affine space associated with

V

. An

affine subspace

of

A

is the

collection

B

of points of

A

that is mapped to a

vector subspace

S

of

V

by the

induced

function

f

⁢

(

P

,

-

)

for some point

P

∈

A

. In other words,

B

is the

inverse image

of

S

under the function

f

⁢

(

P

,

-

)

:

B

=

{

Q

∈

A

∣

f

⁢

(

P

,

Q

)

∈

S

}

.

If

f

is restricted to

B

×

B

, then

B

is an affine space associated with

W

, since

f

⁢

(

Q

,

R

)

=

f

⁢

(

P

,

R

)

-

f

⁢

(

P

,

Q

)

∈

W

, given that

Q

,

R

∈

B

.

For example, if

V

is considered as an affine space associated with

V

with the map

f

⁢

(

v

,

w

)

=

w

-

v

, then an affine subspace

B

of

V

is just a coset of a

subspace

of

V

. In other words,

B

=

S

+

v

, where

S

is a subspace of

V

and

v

∈

V

is a vector. It is evident that

B

is uniquely determined by

S

, and

v

up to

translation

by a vector in

S

. In other words, any two cosets of

S

are affinely isomorphic.

In an affine space

A

, an

affine point

,

affine line

, or

affine plane

is a

0

,

1

, or

2

dimensional affine subspace. Thus, an affine point is just the inverse image of the origin

0

∈

V

. The codimension of an affine subspace is the codimension of the associated vector subspace. An

affine hyperplane

is an affine subspace with codimension 1. When there is no confusion, we may drop the word “affine” in affine point, affine line, etc… Affine subspaces are also called

flats

.

Affine Geometry

Affine geometry

is, generally speaking, the study the geometric properties of affine subspaces. In particular, it is the study of the incidence structure on affine subspaces. Operationally, we may define an

affine geometry

𝒜

⁢

(

V

)

of a vector space

V

to be the poset of all affine subspaces of

V

, orderd by set theoretic

inclusion

. Points in

𝒜

⁢

(

V

)

are commonly written without the set theoretic brackets, so that

v

∈

𝒜

⁢

(

V

)

means

{

v

}

∈

𝒜

⁢

(

V

)

.

Next, we can define an

incidence relation

I

on

𝒜

⁢

(

V

)

so that

(

A

,

B

)

∈

I

iff

A

⊆

B

or

B

⊆

A

. Together with

I

,

𝒜

⁢

(

V

)

becomes an incidence geometry. Two flats

A

and

B

are said to be

parallel

if they have the same associated subspace. As a result, two parallel flats are never

incident

unless they are equal. Also, given a point

v

∈

𝒜

⁢

(

V

)

not incident with

A

, we can always find a flat

B

incident with

v

and parallel to

A

. If

A

=

S

+

w

with

w

≠

v

, simply take

B

=

S

+

v

. This makes

𝒜

⁢

(

V

)

an affine incidence geometry.

In

addition

, we define

A

∨

B

to be the smallest flat in

𝒜

⁢

(

V

)

that contains both

A

and

B

. By Zorn’s lemma,

A

∨

B

exists. Since

A

∨

B

is also unique,

∨

is well-defined. This turns

𝒜

⁢

(

V

)

into an

upper semilattice

. If

S

1

is the associated subspace of

A

and

S

2

is the associated subspace of

B

, then

span

⁡

(

S

1

∪

S

2

)

is the associated subspace of

A

∨

B

. The definition of

∨

can be extended to an arbitrary set of flats, so that

⋁

𝒮

is the smallest flat that contains all flats in

𝒮

⊆

𝒜

. In fact, it is not hard to see that

𝒜

⁢

(

V

)

is

complete semilattice

.

However, since

A

∩

B

may be empty,

𝒜

⁢

(

V

)

is not a

lattice

in general via the “meet” (

=

∩

)

operation

. But when

A

∩

B

≠

∅

,

A

∩

B

∈

𝒜

⁢

(

V

)

. So

∩

is a partially defined operator on

𝒜

⁢

(

V

)

. If one adjoins the

empty set

∅

to

𝒜

⁢

(

V

)

, then

𝒜

⁢

(

V

)

becomes a lattice.

∅

is called the null subspace and its dimension is defined to be

-

1

. One can show that

𝒜

⁢

(

V

)

is a

geometric lattice

.

Although

0

∈

𝒜

⁢

(

V

)

, it is not special, since all points are treated equally; there is no notion of an origin in

𝒜

⁢

(

V

)

. The notion of a metric is also absent, since the underlying vector space is not assumed to have an

inner product

. In fact,

perpendicularity

is not defined in

𝒜

⁢

(

V

)

. In contrast, both notions are important in

Euclidean geometry

, where an inner product has been defined, so that

0

is the unique vector with

0

length.

Affine versus Projective

Affine geometry and

projective geometry

are intimately related. Given an affine geometry

𝒜

⁢

(

V

)

one can construct projective geometries. One easy way is to identify flats that are parallel to each other. Because the parallel

relation

∥

is an

equivalence relation

, we can

partition

𝒜

⁢

(

V

)

into

equivalence classes

. Since each equivalence class is represented by exactly one subspace

S

of

V

, so

𝒜

(

V

)

/

∥

can be identified with

P

⁢

G

⁢

(

V

)

. Of course,

P

⁢

G

⁢

(

V

)

can also be viewed as a sub-poset of

𝒜

⁢

(

V

)

(simply by taking all the subspaces of

V

in

𝒜

⁢

(

V

)

). More generally, if we fix any point

v

∈

𝒜

⁢

(

V

)

, and take all flats that are incident with

v

, the resulting subset

P

⁢

G

v

⁢

(

V

)

forms a

modular

complemented

geometric lattice that is isomorphic to

P

⁢

G

⁢

(

V

)

. In fact,

P

⁢

G

v

⁢

(

V

)

has the

structure

of a projective geometry.

Another way to construct a projective geometry from an affine one is to adjoin extra

elements

to

𝒜

⁢

(

V

)

. Remember that

𝒜

⁢

(

V

)

itself is not a lattice, but simply adjoining

∅

to

𝒜

⁢

(

V

)

won’t give us a projective geometry either, because the resulting lattice is not modular (take two

parallel lines

ℓ

1

,

ℓ

2

and a point

P

lying on

ℓ

1

; then

P

∨

(

ℓ

2

∧

ℓ

1

)

=

P

, while

(

P

∨

ℓ

2

)

∧

ℓ

1

=

ℓ

1

). We start by taking a vector space

U

such that

V

is a subspace of

U

of codimension 1 (This can be done by

linear algebra

). Our objective is to show that

𝒜

⁢

(

V

)

is embeddable in

P

⁢

G

⁢

(

U

)

.

Let

u

∈

U

be a non-zero vector and look at the affine hyperplane

V

+

u

. Each affine subspace of the form

S

+

v

in

V

has the form

S

+

v

+

u

in

V

+

u

, where

S

is a subspace of

V

and

v

∈

V

. Let

𝒜

u

⁢

(

V

)

the collection of all affine subspaces

S

+

v

+

u

(affine subspaces of

U

that are incident with

V

+

u

). There is an obvious one-to-one order preserving correspondence between

𝒜

⁢

(

V

)

and

𝒜

u

⁢

(

V

)

.

Next, every affine subspace

S

+

v

+

u

is the

intersection

of

V

+

u

and a subspace

W

of

U

such that

S

+

v

+

u

⊆

W

and

dim

⁡

(

W

)

=

S

+

1

. Just take

W

=

span

⁡

(

S

,

v

+

u

)

. Clearly

S

+

v

+

u

⊆

W

∩

V

+

u

. In addition,

v

+

u

∉

S

⊆

V

, or else

u

∈

V

gives us a

contradiction

. So

dim

⁡

(

W

)

=

S

+

1

. Finally, if

x

∈

V

+

u

∩

W

, then

x

=

y

+

u

=

s

+

k

⁢

(

v

+

u

)

, where

y

∈

V

,

s

∈

S

, and

k

∈

D

. So

x

≡

u

≡

k

⁢

u

(

mod

V

)

. This implies

(

1

-

k

)

⁢

u

∈

V

. But

u

∉

V

,

k

=

1

. Therefore

x

=

s

+

u

+

v

∈

S

+

v

+

u

. This means that

W

∩

V

+

u

=

S

+

v

+

u

.

The above paragraph shows there is a one-to-one order preserving map from

𝒜

⁢

(

V

)

to

P

⁢

G

⁢

(

U

)

. If we delete all subspaces of

V

from

P

⁢

G

⁢

(

U

)

, and call

P

⁢

G

⁢

(

U

)

/

P

⁢

G

⁢

(

V

)

=

{

W

∈

P

⁢

G

⁢

(

U

)

∣

W

⁢

is not a subspace of

⁢

V

}

,

then we actually get an

order-preserving

bijection between

𝒜

⁢

(

V

)

and

P

⁢

G

⁢

(

U

)

/

P

⁢

G

⁢

(

V

)

.

Title

affine geometry

Canonical name

AffineGeometry

Date of creation

2013-03-22 15:58:20

Last modified on

2013-03-22 15:58:20

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

21

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51A45

Classification

msc 51A15

Synonym

associated subspace

Related topic

AffineCombination

Related topic

AffineTransformation

Defines

affine space

Defines

flat

Defines

associated linear subspace

Defines

affine subspace

Defines

dimension of an affine space

Defines

affine transformation

Defines

affine hyperplane

Defines

affine line

Defines

affine point

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