TRANSFORMATIONS AND PROJECTIONS 29
2.2.4 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
Q
x
y
p
q
p
q
x
y
q
q
*
q
*= q
1
=
10
01
001
cos –sin 0
sin cos 0
001
10–
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*
y
x
y
x
y
x
* 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 P
Q′
P′
P
Q
O x
y
(a)PQ in original position (b) Translating P to P
O x
y
(c) Rotating PQ about the z axis
O x
y
(d) Translating P to its original position
O x
y
Q* Q
P
Q′′ Q′
P′
θ
Q′
θ
P′
Q′′
P
Q