TRANSFORMATIONS AND PROJECTIONS 292.2.4 Rotation of a Point Q (xq,yq, 1) about a Point P (p,q, 1)
Since the rotation matrix Rabout the z-axis and translation matrix T in the x-y plane are known from
Eqs. (2.4) and (2.3) respectively, rotation of QaboutP can be regarded as translating Pto coincide
with the origin, followed by rotation about the z-axis by an angle θ, and lastly, placing P back to its
original position (Figure 2.6). These transformations can be concatenated as
Qx
yp
qp
qx
yq
q
*q
*= q
1=10
01
001cos –sin 0
sin cos 0
00110–
01–
00 1 1⎡ *⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥θθ
θθ (2.7)2.2.5 Reflection
In 2-D, reflection of an object can be obtained by rotating it through 180° about the axis of reflection.
For instance, if an object S in the x-y plane is to be reflected about the x-axis (y = 0), reflection of a
point (x,y, 1) in S is given by (x,y, 1) such that
x*
yx
yx
yx
* fx y
1=–
1=100
0–10
001 1=
1⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥R
(2.8)Similarly, reflection about the y-axis is described as
Figure 2.6 Steps to rotate point Q about point PQ′P′PQO xy(a)PQ in original position (b) Translating P to P
O xy(c) Rotating PQ about the z axisO xy(d) Translating P to its original positionO xy
Q* QPQ′′ Q′P′θQ′θ
P′Q′′PQ