Rotation Transforms in the Plane - Rotation Transforms for Computer Graphics - page 88

+ +

and substituting the identities for cos (θ β) and sin (θ β) we have

x = R ( cos θ cos β − sin θ sin β)

R sin β
= x cos β − y sin β
y = R ( sin θ cos β +

R x

y

= −

R cos β

cos θ sin β)

R y

R sin β

x

=

R cos β

+

= x sin β + y cos β

or in matrix form

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

x
y 1

cos β − sin β 0 x

y
1

⎣ ⎦ = ⎣ ⎦ ⎣ ⎦

sin β cos β 0

0 0 1

and is the homogeneous transform for rotating points about the origin. For example,
to rotate a point 90° about the origin the transform becomes
⎡

⎤ ⎡ ⎤ ⎡ ⎤

0
1
1

0 10

100
001

− 1

0
1

⎣ ⎦ = ⎣ ⎦ ⎣ ⎦

where we see the point ( 1 , 0 , 1 ) becomes ( 0 , 1 , 1 ) and we ignore the homogeneous
scaling factor of 1.
Rotating a point 360° about the origin the transform becomes the identity matrix:
⎡

⎤ ⎡ ⎤ ⎡ ⎤

x
y 1

100
010
001

x
y
1

⎣ ⎦ = ⎣ ⎦ ⎣ ⎦ .

The following observations can be made about the rotation matrix R β :

⎡ ⎤

cos β − sin β 0

⎣ ⎦ .

R β =

sin β cos β 0

0 0 1

Its determinant equals 1:

cos 2 β sin 2 β det R β = + = 1 .

Its transpose is

⎡ ⎤

cos β sin β 0

R β =

⎣ ⎦ .

− sin β cos β 0

0 0 1

The product R β R β = I :

⎡ ⎤ ⎡

⎤
⎦ =

⎡
⎣

⎤
⎦

cos β − sin β 0 cos β sin β 0

100
010
001

⎣ ⎦ ⎣

R β R β =

sin β cos β 0 − sin β cos β 0

0 0 1 0 0 1