Transformations - Transformations and Projections in Computer Graphics - page 42

x x y

y z z

Figure 1.25: Direction of Three-Dimensional Rotations.

counterclockwise rotation. To achieve clockwise rotation, either use a negative angle or
the inverse of the rotation matrix. Inverting our rotation matrices is especially easy and
requires only that we change the signs of the sine functions.
Example . Consider the following compound transformation: (1) a translation by
l , m ,and n units along the three coordinate axes, (2) a rotation of θ degrees about the
x axis, (3) a rotation of φ degrees about the y axis, and (4) the reverse translation. The
four transformation matrices are

⎛ ⎞ ⎛ ⎞

1000
0100
0010
lmn 1

10 00
01 00
00 10
−

⎝ ⎠ ⎝ ⎠

T r = , T rr = ,

l − m − n 1

⎛
⎝

⎞
⎠

⎛
⎝

⎞
⎠

1 0 0 0 cos φ 0 − sin φ 0

0 θ sin θ 0 0 1 0 0

R x = , R y = .

0 − sin θ cos θ 0 sin φ 0 φ 0

0 0 0 1 0 0 0 1

Their product equals the 4 × 4matrix

T = T r R x R y T rr

⎛ ⎞

cos φ 0 − sin φ 0

⎝ ⎠

sin φ sin θ cos θ cos φ sin θ 0

cos θ sin φ − sin θ cos φ cos θ 0

− l + l cos φ

+ m cos( φ

m
+ m cos θ
−

− [ − 2 n + n cos( φ θ )

+ n cos( φ + θ )
−

− 1

= .

− θ ) / 2

− m cos( φ + θ ) / 2

+ n sin( φ

n sin θ 2 l sin φ

θ ) / 2
+ n sin( φ + θ ) / 2

−

θ )
+ m sin( φ + θ )] / 2

− m sin( φ −

Substituting the values θ =30 ◦ , φ =45 ◦ ,and l = m = n = − 1, we get the 4 × 4 matrix

⎛ ⎞

0 . 7071 0 − 0 . 7071 0

⎝ 0 . 3540 0 . 866 0 . 3540 0 ⎠

T = .

0 . 6124 − 0 . 50 0 . 6124 0

− 0 . 673 0 . 634 0 . 7410 1