affine transformation

affine transformation

Definition 1

.

Let

(

A

i

,

f

i

)

be

affine spaces

associated with a left (right)

vector spaces

V

i

(over some

division ring

D

), where

i

=

1

,

2

. An

affine transformation

from

A

1

to

A

2

is a

function

α

:

A

1

→

A

2

such that there is a

linear transformation

T

:

V

1

→

V

2

such that

T

⁢

(

f

1

⁢

(

P

,

Q

)

)

=

f

2

⁢

(

α

⁢

(

P

)

,

α

⁢

(

Q

)

)

for any

P

,

Q

∈

A

.

Note that

T

is uniquely determined by

α

, since

f

1

is a function onto

V

1

.

T

and is called the

associated linear transformation

of

α

. Let us write

[

α

]

the associated linear transformation of

α

. Then the definition above can be illustrated by the following

commutative diagram

:

\xymatrix

⁢

@

+

=

2

⁢

c

⁢

m

⁢

A

1

×

A

1

⁢

\ar

⁢

[

r

]

-

⁢

f

1

⁢

\ar

⁢

[

d

]

(

α

,

α

)

⁢

&

⁢

V

1

⁢

\ar

⁢

[

d

]

[

α

]

⁢

A

2

×

A

2

⁢

\ar

⁢

[

r

]

-

⁢

f

2

⁢

&

⁢

V

2

Here’s an example of an affine transformation. Let

(

A

,

f

)

be an affine space with

V

the associated vector space. Fix

v

∈

V

. For each

P

∈

A

, let

α

⁢

(

P

)

be the unique point in

A

such that

f

⁢

(

P

,

α

⁢

(

P

)

)

=

v

. Then

α

:

A

→

A

is a well-defined function. Furthermore,

f

⁢

(

α

⁢

(

P

)

,

α

⁢

(

Q

)

)

=

v

+

f

⁢

(

α

⁢

(

P

)

,

α

⁢

(

Q

)

)

-

v

=

f

⁢

(

P

,

α

⁢

(

P

)

)

+

f

⁢

(

α

⁢

(

P

)

,

α

⁢

(

Q

)

)

+

f

⁢

(

α

⁢

(

Q

)

,

Q

)

=

f

⁢

(

P

,

Q

)

=

1

V

⁢

(

f

⁢

(

P

,

Q

)

)

. Thus

α

is affine, with

[

α

]

=

1

V

.

An affine transformation

α

:

A

1

→

A

2

is an

affine isomorphism

if there is an affine transformation

β

:

A

2

→

A

1

such that

β

∘

α

=

1

A

1

and

α

∘

β

=

1

A

2

. Two affine spaces

A

1

and

A

2

are

affinely isomorphic

, or simply,

isomorphic

, if there are affine isomorphism

α

:

A

1

→

A

2

.

Below are some basic properties of an affine transformation (see proofs here (

http://planetmath.org/PropertiesOfAnAffineTransformation

)):

1.

α

is onto iff

[

α

]

is.

2.

α

is

one-to-one

iff

[

α

]

is.

3.

A

bijective

affine transformation

α

is an affine isomorphism. In fact,

[

α

-

1

]

=

[

α

]

-

1

.

4.

Two affine spaces associated with the same vector space are isomorphic.

Because of the last property, it is often enough, in practice, to identify

V

itself as

the

affine space associated with

V

, up to affine isomorphism, with the

direction

given by

f

⁢

(

v

,

w

)

=

w

-

v

. With this in mind, we may reformulate the definition of an affine transformation as a

mapping

α

from one vector space

V

to another,

W

, such that there is a linear transformation

T

:

V

→

W

such that

T

⁢

(

w

-

v

)

=

α

⁢

(

w

)

-

α

⁢

(

v

)

.

By fixing

w

∈

V

, we get the following

equation

α

⁢

(

v

)

=

T

⁢

(

v

)

+

(

α

⁢

(

w

)

-

T

⁢

(

w

)

)

.

Definition 2

.

Let

V

and

W

be left vector spaces over the same division ring

D

. An

affine transformation

is a mapping

α

:

:

V

→

W

such that

α

⁢

(

v

)

=

T

⁢

(

v

)

+

w

,

v

∈

V

for some linear transformation

T

:

V

→

W

and some vector

w

∈

W

.

An

affine property

is a

geometry

property that is preserved by an affine transformation. The following are affine properties of an affine transformation Let

A

:

V

→

W

:

•

linearity. Given an affine subspace

S

+

v

of

V

, then

A

⁢

(

S

+

v

)

=

L

⁢

(

S

+

v

)

+

w

=

L

⁢

(

S

)

+

(

L

⁢

(

v

)

+

w

)

is an affine subspace of

W

.

•

incidence. Suppose

S

+

v

⊆

T

+

u

. Pick

x

∈

A

⁢

(

S

+

v

)

=

L

⁢

(

S

)

+

L

⁢

(

v

)

+

w

, so

x

=

y

+

L

⁢

(

v

)

+

w

where

y

∈

L

⁢

(

S

)

. Since

L

is bijective, there is

z

∈

S

such that

L

⁢

(

z

)

=

y

. So

A

⁢

(

z

+

v

)

=

L

⁢

(

z

)

+

L

⁢

(

v

)

+

w

=

x

. Since

z

+

v

∈

S

+

v

,

z

+

v

=

t

+

u

for some

t

∈

T

,

x

=

A

⁢

(

z

+

v

)

=

A

⁢

(

t

+

u

)

∈

A

⁢

(

T

+

u

)

. Therefore,

A

⁢

(

S

+

v

)

⊆

A

⁢

(

T

+

u

)

.

•

parallelism

. Given two

parallel

affine subspaces

S

+

a

and

S

+

b

, then

A

⁢

(

S

+

a

)

=

L

⁢

(

S

)

+

(

L

⁢

(

a

)

+

w

)

and

A

⁢

(

S

+

b

)

=

L

⁢

(

S

)

+

(

L

⁢

(

b

)

+

w

)

are parallel.

•

coefficients of an

affine combination

. Given that

v

is an affine combination of

v

1

,

…

,

v

n

:

v

=

k

1

⁢

v

1

+

⋯

+

k

n

⁢

v

n

,

where

k

i

∈

F

⁢

and

⁢

k

1

+

⋯

+

k

n

=

1

are the corresponding coefficients. Then

A

⁢

(

v

)

=

k

1

⁢

L

⁢

(

v

1

)

+

⋯

+

k

n

⁢

L

⁢

(

v

n

)

+

w

=

k

1

⁢

(

L

⁢

(

v

1

)

+

w

)

+

⋯

+

k

n

⁢

(

L

⁢

(

v

n

)

+

w

)

=

k

1

⁢

A

⁢

(

v

1

)

+

⋯

+

k

n

⁢

A

⁢

(

v

n

)

is the affine combination of

A

⁢

(

v

1

)

,

…

,

A

⁢

(

v

n

)

with the same set of coefficients.

Special Affine Transformations

1.

translation

.
An affine transformation of the form

A

⁢

(

v

)

=

v

+

w

is called a

translation

. Every affine transformation can be decomposed as a

product

of a linear transformation and a translation:

A

⁢

(

v

)

=

L

⁢

(

v

)

+

w

=

B

⁢

C

⁢

(

v

)

where

C

⁢

(

v

)

=

L

⁢

(

v

)

and

B

⁢

(

v

)

=

v

+

w

. The order of

composition

is important, since

B

⁢

C

≠

C

⁢

B

. Geometrically, a translation moves a geometric figure along a

straight line

.

2.

dilation

(map)

.
If

L

has a unique

eigenvalue

d

≠

0

(that is,

L

may be diagonalized as

d

⁢

I

, the

diagonal matrix

with non-zero

diagonal

entries

=

d

∈

F

), then the affine transformation

A

⁢

(

v

)

=

L

⁢

(

v

)

is called a

dilation

. Note that a dilation may be written as the product of a vector with a scalar:

A

⁢

(

v

)

=

d

⁢

v

, which is why a dilation is also called a

scaling

. A dilation can be visualized as magnifying or shrinking a geometric figure.

3.

homothetic transformation

.
The composition of a dilation followed by a translation is called a

homothetic transformation

. It has the form

A

⁢

(

v

)

=

d

⁢

v

+

w

,

0

≠

d

∈

F

.

4.

Euclidean transformation

. In the case when both

V

and

W

are

Euclidean vector spaces

, if the associated linear transformation is

orthogonal

, then the affine transformation is called a

Euclidean transformation

.

Remarks

1.

When

V

=

W

, the set of affine maps

V

→

V

, with function composition as the product, becomes a group, and is denoted by

IGL

⁢

(

V

)

. The

multiplicative identity

is the

identity map

. If

A

⁢

(

v

)

=

L

⁢

(

v

)

+

w

, then

A

-

1

⁢

(

v

)

=

L

-

1

⁢

(

v

)

-

L

-

1

⁢

(

w

)

. IGL is short for

of

V

. Translations, dilations, and homothetic transformations all form

subgroups

of

IGL

⁢

(

V

)

. If

T

is the group of translations,

D

the group of dilations, and

H

the group of homothetic transformations, then

T

is a

normal subgroup

of

T

. Also,

Aut

⁡

(

T

)

and

Aut

⁡

(

D

)

are

abelian groups

(remember:

F

is assumed to be a field).

2.

One can more generally define an affine transformation to be an order-preserving

bijection

between two

affine geometries

. It can be shown that this definition coincides with the above one if the underlying field admits no non-trivial

automorphisms

. When the two affine geometries are the same, the bijective affine transformation is called an

affinity

.

3.

Another way to generalize an affine transformation is to remove the

restriction

on the invertibility of the linear transformation

L

. In this respect, the set

A

⁢

(

V

,

W

)

of affine transformations from

V

to

W

has a natural vector space

structure

. It is easy to see that the set

L

⁢

(

V

,

W

)

of linear transformations from

V

to

W

forms a

subspace

of

A

⁢

(

V

,

W

)

.

Title

affine transformation

Canonical name

AffineTransformation

Date of creation

2013-03-22 14:46:08

Last modified on

2013-03-22 14:46:08

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

37

Author

matte (1858)

Entry type

Definition

Classification

msc 51A10

Classification

msc 51A15

Classification

msc 15A04

Synonym

scaling

Related topic

LinearTransformation

Related topic

AffineSpace

Related topic

ComplexLine

Related topic

AffineCombination

Related topic

AffineGeometry

Related topic

Collineation

Defines

IGL

Defines

translation

Defines

dilation

Defines

dilation map

Defines

homothetic transformation

Defines

affine property

Defines

affine isomorphism

Defines

associated linear transformation

Defines

affinely isomorphic

Defines

affinity

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