Transformation - Mathematics for Computer Graphics - page 83

Therefore

T 1 rotates φ about the y -axis
T 2 rotates

θ about the z -axis
T 3 rotates α about the x -axis
T 4 rotates θ about the z -axis
T 5 rotates

−

− φ about the y -axis

where

⎡ ⎤ ⎡ ⎤

cos φ 0 sin φ cos θ sin θ 0

⎣ ⎦ ⎣ ⎦

T 1 = 0 1 0 T 2 = − sin θ cos θ 0

− sin φ 0 s φ 0 0 1

⎡ ⎤ ⎡ ⎤

10 0 cos θ − sin θ 0

⎣ ⎦ ⎣ ⎦

T 3 = 0 s α − sin α T 4 = sin θ cos θ 0

0 sin α cos α 0 0 1

⎡ ⎤

cos φ sin φ

01 0
sin φ

0 −

⎣ ⎦

T 5 =

0c s φ

Let

⎡ ⎤

E 1 , 1 E 1 , 2 E 1 , 3

⎣ ⎦

T 1 × T 2 × T 3 × T 4 × T 5 = E 2 , 1 E 2 , 2 E 2 , 3

E 3 , 1 E 3 , 2 E 3 , 3

From Figure 7.25

cos θ = 1 cos 2 θ =1 b 2 − b 2 ⇒ −

sin 2 θ = b 2 sin θ = b ⇒

a 2

a

cos 2 φ =

√ 1

cos φ =

b 2 ⇒

1 − b 2

−

c 2 c

sin 2 φ =

√ 1

sin φ =

b 2 ⇒

b 2

1 −

−

To find E 1 , 1

E 1 , 1 =cos 2 φ cos 2 θ +cos 2 φ sin 2 θ cos α +sin 2 φ cos α

· 1 b 2 +

a 2 a 2 c 2

b 2 cos α +

E 1 , 1 = − ·

b 2 cos α 1 − b 2 1 − b 2 1 −

a 2 b 2
1

c 2

E 1 , 1 = a 2 +

b 2 cos α + b 2 cos α

− 1 −

E 1 , 1 = a 2 + c 2 + a 2 b 2
1

cos α

− b 2