TRANSFORMATIONS AND PROJECTIONS 27where
R =
cos – sin 0
sin cos 0
001θθ
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 11
= , 1
1=cos –sin 0
sin cos 0
0011x
yx
y⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥θθ
θθand PP 21
2
21
= , 1
1=10
01
0011=10
01
001cos –sin 0
sin cos 0
0011Tx
yp
qx
yp
qx
y⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥θθ
θθ ⎥⎥
⎥= TRPThus,P 2 =
cos –sin
sin cos
0011θθ
θθp
qx
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 *= = RTPcos –sin 0
sin cos 0
00110
01
001 1=cos – sin cos – sin
sin cos sin + sin
00 1 1θθ
θθθθ θ θ
θθ θ θ⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢p ⎤
qx
ypq
pqx
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.