Euclidean transformation

Euclidean transformation

Let

V

and

W

be

Euclidean vector spaces

. A

Euclidean
transformation

is an

affine transformation

E

:

V

→

W

, given by

E

⁢

(

v

)

=

L

⁢

(

v

)

+

w

such that

L

is an

orthogonal

linear
transformation

(

http://planetmath.org/OrthogonalTransformation

).

As an affine transformation, all affine properties, such as
incidence and

parallelism

are preserved by

E

. In

addition

, since

E

⁢

(

u

-

v

)

=

L

⁢

(

u

-

v

)

and

L

is an ,

E

preserves

lengths of

line segments

and

angles between two line

segments (

http://planetmath.org/AngleBetweenTwoLines

). Because of this, a Euclidean transformation is also called
a

rigid motion

, which is a popular term used in mechanics.

Types of Euclidean transformations

There are three main
types of Euclidean transformations:

1.

translation

. If

L

=

I

, then

E

is just a translation. Any
Euclidean transformation can be decomposed into a

product

of an

orthogonal transformation

L

⁢

(

v

)

, followed by a
translation

T

⁢

(

v

)

=

v

+

w

.

2.

rotation

. If

w

=

0

, then

E

is just an orthogonal transformation. If

det

⁡

(

E

)

=

1

, then

E

is called a

rotation

. The

orientation

of any basis (of

V

) is preserved under a rotation. In the
case where

V

is two-dimensional, a rotation is

conjugate

to a matrix of the form

(

cos

⁡

θ

-

sin

⁡

θ

sin

⁡

θ

cos

⁡

θ

)

,

(1)

where

θ

∈

ℝ

. Via this particular (unconjugated) map, any vector emanating from the origin is rotated
in the counterclockwise

direction

by an angle of

θ

to another vector emanating from the origin. Thus, if

E

is conjugate to the matrix given above, then

θ

is the

angle of rotation

for

E

.

3.

reflection

. If

w

=

0

but

det

⁡

(

E

)

=

-

1

instead, then

E

is a called

reflection

. Again, in the two-dimensional case, a reflection is
to a matrix of the form

(

cos

⁡

θ

sin

⁡

θ

sin

⁡

θ

-

cos

⁡

θ

)

,

(2)

where

θ

∈

ℝ

. Any vector is reflected by this particular (unconjugated) map to another
by a “mirror”, a line of the form

y

=

x

⁢

tan

⁡

(

θ

2

)

.

Remarks

.

•

In general, an orthogonal transformation can be represented by a matrix of the
form

(

A

1

O

⋯

O

O

A

2

⋯

O

⋮

⋮

⋱

⋮

O

O

⋯

A

n

)

,

where each

A

i

is either

±

1

or a

rotation matrix

(1) (or reflection
matrix (2)) given above. When its

determinant

is -1 (a reflection), it has an

invariant subspace

of

V

of

codimension

1. One can think of this

hyperplane

as the mirror.

•

Another common rigid motion is the

glide reflection

. It is a Euclidean transformation that is expressible as a product of a reflection, followed by a translation.

Title

Euclidean transformation

Canonical

name

EuclideanTransformation

Date of creation

2013-03-22 15:59:46

Last modified on

2013-03-22 15:59:46

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

17

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51A10

Classification

msc 15A04

Classification

msc 51A15

Synonym

rigid motion

Defines

translation

Defines

translate

Defines

rotation

Defines

rotate

Defines

reflection

Defines

reflect

Defines

reflexion

Defines

glide reflection

Defines

angle of rotation

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