Computer Aided Engineering Design

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

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0001

10 0 0

0 1

2

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2

0

0 1

2

1

2

0

00 0 1

1

2

1

2

00

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2

1

2

00

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0001

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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

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⎢

⎢

⎢

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⎥

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(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