Rotation Transforms in the Plane - Rotation Transforms for Computer Graphics - page 93

Similarly, we know that the rotation transform is given by

⎡ ⎤

cos β − sin β 0

⎣ ⎦

R β = sin β cos β 0

0 0 1

and we can reason that the inverse of R β must be a rotation in the opposite direction,
i.e. a rotation of

− β :

⎡ ⎤

cos β sin β 0

R − 1
β

⎣ ⎦ .

= − sin β cos β 0

0 0 1

We can also compute R − β by forming the cofactor matrix of R β , transposing it and
dividing by its determinant:

⎡ ⎤

cos β − sin β 0

⎣ ⎦

cofactor matrix of R β = sin β cos β 0

0 0 1

⎡ ⎤

cos β sin β 0

R β =

⎣ ⎦

− sin β cos β 0

0 0 1

and as det R β = 1, we can write

⎡ ⎤

cos β sin β 0

R − 1
β

⎣ ⎦ .

= − sin β cos β 0

0 0 1

So our reasoning is correct. Furthermore, R β R − 1 = I :

β

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

cos β − sin β 0 cos β sin β 0 100

010
001

R β R − 1

⎣ ⎦ ⎣ ⎦ = ⎣ ⎦ .

= sin β cos β 0 − sin β cos β 0

β

0 0 1 0 0 1

7.5 Multivector Transforms

Multivectors are linear combinations of vectors, bivectors, trivectors, etc., plus a
scalar. They possess imaginary qualities and consequently have the ability to rotate
vectors. Although it is unusual to employ multivectors in 2D computer graphics,
they have been included to introduce their rotational qualities.

7.5.1 Translate a Point

Figure 7.3 shows a point P(x,y) with position vector p , and is translated by (t x ,t y )
using

p = p + t