38 COMPUTER AIDED ENGINEERING DESIGN
L L
z x
y
(a) (b)
Figure 2.14 Rotation of an object: (a) about the line y–x = 0 and (b) rotated result
R = Rz(45°)Rx(45°)Rz(− 45 °)
1
2
-^1
2
00
1
2
1
2
00
0010
0001
10 0 0
0 1
2
-^1
2
0
0 1
2
1
2
0
00 0 1
1
2
1
2
00
-^1
2
1
2
00
0010
0001
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎢⎢
⎢
⎢
⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
and the result is shown in Figure 2.14 (b). An alternate way is to rotate the line about the z-axis to
coincide with the y-axis, perform rotation about the y-axis and then rotate L back to its original
location. Apparently, transformation procedures may not be unique though the end result would be
the same if a proper transformation order is followed.
To rotate a point P about an axis L having direction cosines n = [nxnynz 0] that passes through a
pointA [pqr 1], we may observe that P and its new location P* would lie on a plane perpendicular
toL and the plane would intersect LatQ (Figure 2.15(a)). Transformations may be composed
stepwise as follows:
(i) Point A on L may be translated to coincide with the origin O using the transformation TA. The
new line L′ remains parallel to L.
TA=
100–
010–
001 –
000 1
p
q
r
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
(ii) The unit vector OU (along L) projected onto the x-y and y-z planes, makes the traces OUxy and
OUyz, respectively (Figure 2.15(b)). The magnitude of OUyz is d = √√(nn^22 yz+ ) = (1 – )n^2 x.OUyz
makes an angle ψ with the z-axis such that