determining rotations and reflections in ℝ^2

determining rotations and reflections in

ℝ

2

Let

E

:

ℝ

2

→

ℝ

2

be a

rotation

about some point

(

x

0

,

y

0

)

, and let

θ

the angle of rotation for

E

. A formula for

E

can be determined as follows.

First,

translate

the point

(

x

0

,

y

0

)

to

(

0

,

0

)

. The map of this

translation

is

T

⁢

(

x

,

y

)

=

(

x

-

x

0

,

y

-

y

0

)

.

Next, rotate by

θ

about the origin. The map of this rotation is

R

⁢

(

x

,

y

)

=

(

x

⁢

cos

⁡

θ

-

y

⁢

sin

⁡

θ

,

x

⁢

sin

⁡

θ

+

y

⁢

cos

⁡

θ

)

.

Finally, translate the point

(

0

,

0

)

back to

(

x

0

,

y

0

)

. The map of this translation is

T

-

1

⁢

(

x

,

y

)

=

(

x

+

x

0

,

y

+

y

0

)

.

The fact that

E

=

T

-

1

∘

R

∘

T

can be used to obtain a formula for

E

:

E

⁢

(

x

,

y

)

=

(

T

-

1

∘

R

∘

T

)

⁢

(

x

,

y

)

=

(

T

-

1

∘

R

)

⁢

(

x

-

x

1

,

y

-

y

0

)

=

T

-

1

⁢

(

(

x

-

x

0

)

⁢

cos

⁡

θ

-

(

y

-

y

0

)

⁢

sin

⁡

θ

,

(

x

-

x

0

)

⁢

sin

⁡

θ

+

(

y

-

y

0

)

⁢

cos

⁡

θ

)

=

(

(

x

-

x

0

)

⁢

cos

⁡

θ

-

(

y

-

y

0

)

⁢

sin

⁡

θ

+

x

0

,

(

x

-

x

0

)

⁢

sin

⁡

θ

+

(

y

-

y

0

)

⁢

cos

⁡

θ

+

y

0

)

.

Let

E

:

ℝ

2

→

ℝ

2

be a

reflection

about some line

y

=

m

⁢

x

+

b

. Let

θ

=

2

⁢

arctan

⁡

m

. A formula for

E

can be determined as follows.

First, translate the

y

intercept

(

0

,

b

)

to

(

0

,

0

)

. The map of this translation is

B

⁢

(

x

,

y

)

=

(

x

,

y

-

b

)

.

Next, reflect about the line

y

=

m

⁢

x

. The map of this reflection is

F

⁢

(

x

,

y

)

=

(

x

⁢

cos

⁡

θ

+

y

⁢

sin

⁡

θ

,

x

⁢

sin

⁡

θ

-

y

⁢

cos

⁡

θ

)

.

Finally, translate the point

(

0

,

0

)

back to

(

0

,

b

)

. The map of this translation is

B

-

1

⁢

(

x

,

y

)

=

(

x

,

y

+

b

)

.

The fact that

E

=

B

-

1

∘

F

∘

B

can be used to obtain a formula for

E

:

E

⁢

(

x

,

y

)

=

(

B

-

1

∘

F

∘

B

)

⁢

(

x

,

y

)

=

(

B

-

1

∘

F

)

⁢

(

x

,

y

-

b

)

=

B

-

1

⁢

(

x

⁢

cos

⁡

θ

+

(

y

-

b

)

⁢

sin

⁡

θ

,

x

⁢

sin

⁡

θ

-

(

y

-

b

)

⁢

cos

⁡

θ

)

=

(

x

⁢

cos

⁡

θ

+

(

y

-

b

)

⁢

sin

⁡

θ

,

x

⁢

sin

⁡

θ

-

(

y

-

b

)

⁢

cos

⁡

θ

+

b

)

.

Let

E

:

ℝ

2

→

ℝ

2

be a reflection about some line

x

=

c

. A formula for

E

can be determined as follows.

First, translate the

x

intercept

(

c

,

0

)

to

(

0

,

0

)

. The map of this translation is

C

⁢

(

x

,

y

)

=

(

x

-

c

,

y

)

.

Next, reflect about the line

x

=

0

. The map of this reflection is

M

⁢

(

x

,

y

)

=

(

-

x

,

y

)

.

Finally, translate the point

(

0

,

0

)

back to

(

c

,

0

)

. The map of this translation is

C

-

1

⁢

(

x

,

y

)

=

(

x

+

c

,

y

)

.

The fact that

E

=

C

-

1

∘

M

∘

C

can be used to obtain a formula for

E

:

E

⁢

(

x

,

y

)

=

(

C

-

1

∘

M

∘

C

)

⁢

(

x

,

y

)

=

(

C

-

1

∘

M

)

⁢

(

x

-

c

,

y

)

=

C

-

1

⁢

(

c

-

x

,

y

)

=

(

2

⁢

c

-

x

,

y

)

.

Title

determining rotations and reflections in

ℝ

2

Canonical name

DeterminingRotationsAndReflectionsInmathbbR2

Date of creation

2013-03-22 17:14:34

Last modified on

2013-03-22 17:14:34

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

4

Author

Wkbj79 (1863)

Entry type

Derivation

Classification

msc 51A15

Classification

msc 51A10

Classification

msc 15A04

Related topic

Symmetry2

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