Frames of Reference in Space - Rotation Transforms for Computer Graphics - page 146

Fig. 10.3 ( a )and( b ) The cube is rotated − 90°. ( c )and( d )The XYZ frame is rotated 90°

The transform for R − 1
90 ° ,x

is

⎡ ⎤

100
001
0

R − 1

⎣ ⎦

90 ° ,x =

R − 90 ° ,x =

− 10

which when used on the cube's coordinates create
⎡

⎤ ⎡ ⎤

100
001
0

00001111
00110011
01010101

⎣ ⎦ ⎣ ⎦

−

10

⎡ ⎤

00 0 0 11 1 1

⎣ ⎦

= 01 0 1 01 0 1

00 − 1 − 100 − 1 − 1

which are confirmed by Fig. 10.3 (d).
In summary, the transforms for rotating frames α about the x -, y - and z -axes are:

⎡ ⎤

1 0 0

R − 1

⎣ ⎦

α,x =

0 cos α sin α

0 − sin α cos α

⎡ ⎤

cos α sin α

01 0
sin α

0 −

⎣ ⎦

R − 1

α,y =

0 cos α