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)