TRANSFORMATIONS AND PROJECTIONS 27
where
R =
cos – sin 0
sin cos 0
001
θθ
θθ
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
(2.4)
Rigid body translation and rotation thus get unified as matrix multiplication operations only,
involving no addition or subtraction of matrices and vectors. Further, one can concatenate a sequence
of transformations, for instance, translation of an object followed by its rotation. If one can identify
the matrices for each of these transformations in the multiplication form, it becomes much easier to
track the intermediate positions as well as to predict the final transformed position of the rigid body.
2.2.3 Combined Rotation and Translation
Consider a point P (x,y, 1) in the x-y plane to be rotated by an angle θ about the z-axis to a position
P 1 (x 1 ,y 1 , 1) followed by a translation by v(p,q) to a position P 2 (x 2 ,y 2 , 1). Using Eqs. (2.3) and (2.4),
we may write
PRP 1
1
= , 1
1
=
cos –sin 0
sin cos 0
0011
x
y
x
y
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
θθ
θθ
and PP 21
2
2
1
= , 1
1
=
10
01
0011
=
10
01
001
cos –sin 0
sin cos 0
0011
T
x
y
p
q
x
y
p
q
x
y
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
θθ
θθ ⎥⎥
⎥
= TRP
Thus,P 2 =
cos –sin
sin cos
0011
θθ
θθ
p
q
x
y
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
(2.5)
On the contrary, if translation by v is followed by rotation about the z-axis by an angle θ to reach P 2 *,
then
P 2 *= = RTP
cos –sin 0
sin cos 0
001
10
01
001 1
=
cos – sin cos – sin
sin cos sin + sin
00 1 1
θθ
θθ
θθ θ θ
θθ θ θ
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
p ⎤
q
x
y
pq
pq
x
y
⎦⎦
⎥
⎥
⎥
(2.6)
We observe from Eqs. (2.5) and (2.6) that the final positions P 2 and P 2 * are not identical. From above
we can arrive at two important conclusions: (a) the homogeneous coordinate system helps to unify
translation and rotation as multiplicative transformations and (b) transformations are not commutative.
The sequence in which the transformations are performed is significant and must be maintained while
concatenating the respective matrices. Otherwise a different orientation or position of the object is
reached. If T 1 ,T 2 , ..., Tn are the transformations to be performed in the order, the combined transformation
matrixTis given as T = Tn Tn– 1 Tn– 2 ... T 2 T 1.