Computer Aided Engineering Design

TRANSFORMATIONS AND PROJECTIONS 39

y

x

O

U

U′ Uyz

z

φ

d

ψ

Figure 2.15(b) Computing angles from the direction

cosines

cos = ψψ, sin =

n

d

n

d

z y

RotateOU about the x-axis by ψ to place it on the x-z plane (OU′) in which case OUyzwill coincide
with the z-axis.OU′ makes angle φ with the z-axis such that cos φ = d and sin φ = nx. Rotate OU′ about
they-axis by –φ so that in effect, OU coincides with the z-axis. The two rotation transformations are
given by

RRxy=

10 0 0

0 cos –sin 0

0 sin cos 0

00 0 1

and =

cos(– ) 0 sin(– ) 0

0100

–sin(– ) 0 cos(– ) 0

0001

ψψ

ψψ

φφ

φφ

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

(iii) The required rotation through angle α is then performed about the z-axis using

Rz=

cos –sin 0 0

sin cos 0 0

0010

0001

αα

αα

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

(iv) Eventually, OU or line L is placed back to its original location by performing inverse transformations.
The complete rotation transformation of point P about L can now be written as

R = TA–1 Rx–1(ψ)Ry–1(−φ)Rz(α) Ry(−φ)Rx(ψ)TA (2.28)

Figure 2.16 shows, as an example, the rotation of a disc about its axis placed arbitrarily in the
coordinate system. Note that all matrices being orthogonal, Ry–1(–φ) = Ry(φ),Rx–1(ψ)=Rx (– ψ) and

TA–1(–v) = TA(v), where v = [pqr]T.

Figure 2.15(a) Rotation of P about a line L

z

O

x

y

A(p,q,r,s)

Q

P

P*

L (nx,ny,nz, 0)