TRANSFORMATIONS AND PROJECTIONS 25are most convenient to represent such motions. The homogenous coordinate system, which has some
distinct advantages, is also introduced to unify the two transformations.
2.2.1 Rotation in Two-Dimensions
Consider a rigid body S packed with points Pi (i = 1, ..., n) and let a point Pj(xj,yj) on S be rotated
about the z-axis to Pjjj***(, )xy by an angle θ. From Figure 2.3, it can be observed that
transformation in Eq. (2.1) must be performed simultaneously for all points Pi(i = 1,... , n) such
that the entire rigid body reaches the new destination S*.
Example 2.1 A trapezoidal lamina ABCD lies in the x-y plane as shown with A(6, 1), B(8, 1),
C(10, 4) and D(3, 4). The lamina is to be rotated about the z-axis by 90°. Determine the new position
ABCD after rotation (Figure 2.4(a)).
x*j = l cos (θ +α)=l cos α cos θ−lsinα sin θ= xjcosθ−yjsinθandyj* = lsin (θ + α)=lcosα sin θ+lsinα cos θ
=xjsinθ + yj cos θ
Or in matrix formx
yx
yj
jj
jjj*
*
= cos – sin *
sin cos=⎡
⎣⎢⎤
⎦⎥⎡
⎣⎢⎤
⎦⎥⎡
⎣⎢⎤
⎦⎥⇒θθ
θθPRP (2.1)where R =
cos – sin
sin cosθθ
θθ⎡
⎣
⎢⎤
⎦
⎥ is the two-dimensionalrotation matrix. For S to be rotated by an angle θ,
Figure 2.3 Rotation in a planelxjPj(xj, yj)lαθOyjy*jxj*P,jjj***( )xyThe transformation matrix R is given by Eq. (2.1)
withθ = 90°. Thus,A*
B
C
DA
B
C
DT T T*
*
*= =cos 90 –sin 90
sin 90 cos 9061
81
10 4
34⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥°°
°°⎡
⎣⎢⎤
⎦⎥⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥R=0–1
1061
81
10 4
34=–1 6
–1 8
–4 10
–4 3⎡
⎣
⎢⎤
⎦
⎥⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥TT2.2.2 Translation in Two-Dimensions: Homogeneous Coordinates
For a rigid body S to be translated along a vector v such that each point of S shifts by (p,q),
xxpyyqx
yx
yp
q
j
*
jjjj
jj
j`= + , = + = + = + PPvjj
**
*
⇒⎡ *
⎣⎢
⎢⎤⎦⎥
⎥⎡⎣⎢⎤⎦⎥⎡
⎣⎢⎤
⎦⎥⇒ (2.2)Figure 2.4 (a) Lamina rotation in Example 2.1D CA BD*A*B*C*- 4 –2 0 2 4 6 8 10
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