Rotation Matrices - University of Utah

4.

c)

Rotation Matrices

vi

0

0

Suppose that 2 R. We let

C

vi

vi

0 0

4.

R : R2 ! R2

be the function defined as follows: Any vector in the plane can be written in polar coordinates as r(cos(), sin())

where r 0 and 2 R. For any such vector, we define

C

vi

R r( cos(), sin() ) = r( cos( + ), sin( + ) )

Notice that the function R doesn't change the norms of vectors (the num-

ber r), it just aects their direction, which is measured by the unit circle

c)

coordinate.

We call the function R

.

rotation of the plane by angle

+

vi

+

vi

0 0

C

vi

4.

c)

+ +C

vi

+

0

0

A

vi 0

+

+

vi

0 0

4.

C

vi

c)

+

vi

+

0

+

v-b

0 0

If > 0, then R rotates the plane counterclockwise by an angle of . If

< 0, then R is a clockwise rotation by an angle of ||. The rotation does

not aect the origin in the plane. That is, R (0, 0) = (0, 0) always, no matter

which number is.

258

+

+

0

Examples.

? Because

R

2

is the

2

> 0, it

function that rotates the plane by an angle of is a counterclockwise rotation. Thus, R (1, 1) is

2

2

,

or

90.

the point

in the plane that we obtain by rotating (1, 1) counterclockwise by an angle

of

2

.

I

--

(H) a

`

S

in

the

?

Because

-

2

< 0,

R-

2

is

a

clockwise

rotation.

R-

2

(1,

plane obtained by rotating (1, 1) clockwise by an angle

1) of

is the

2

.

point

(I,')

? The function R0 : R2 R2 rotates the plane by an angle of 0. That is, it doesn't rotate the plane at all. It's just the identity function for the plane.

259

? Below is the picture of a shape in the plane. It's a triangle, and we'll call this subset of the plane D.

D

R (D) is the set in the plane obtained by rotating D counterclockwise by

an

2

angle

of

2

.

(It's

counterclockwise

because

2

>

0.)

R-

4

(D)

is

D

rotated

clockwise

by

an

angle

of

4

.

260

? Let's rotate the vector (a, 0), where a 0. This is a point on the x-axis whose norm equals a.

I (i,o) (ao'):o.(l,o)

We can write this vector in polar coordinates as a(1, 0), or equivalently, as a( cos(0), sin(0) ). Now we can rotate the vector (a, 0) by an angle . That's the vector R(a, 0), which by the formula from the beginning of this chapter is

R(a, 0) = R a( cos(0), sin(0) ) = a( cos(0 + ), sin(0 + ) ) = a( cos(), sin() )

(o'i)v: (o'z) `? (o'z) `?

(o'i)v: (o'i)v:

0 0

(o'z) `? (o'z) `?

(o'i)v: (o'i)v:

? In this example, we'll rotate a vector (0, b), where b 0. This is a

vector whose norm equals b, and that points straight up. In polar coordinates,

(0, b) = b( cos

2

, sin

2

).

(o'z) `?

0

(o,)

fir "a

261

Now if we rotate (0, b) by an angle , then we have

R(0, b) = R

b

cos

, sin 2

2

= b cos + , sin +

2

2

= b cos + , sin +

2

2

There's a slightly better way to write the result above, but it requires a cou-

ple of the identities we learned in the chapter "Sine and Cosine". Specifically,

Lemmas 8-10 tell us that

cos +

2

= cos + -

2 = sin( + )

= - sin()

and

sin + = cos()

2

Therefore,

R(0, b) = b

cos

+ 2

, sin

+ 2

[Lemma 9] [Lemma 10]

[Lemma 8] = b( - sin(), cos() )

(o,)

cos (oc))

*************

262

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