Rotation Transforms in Space - Rotation Transforms for Computer Graphics - page 135

work, it is nevertheless extremely tedious, but is a good exercise for improving one's
algebraic skills!
Figure 9.10 shows a point P(x,y,z) to be rotated through an angle α to
P (x ,y ,z ) about an axis defined by
ˆ c k .

The transforms to achieve this operation can be expressed as follows:
⎡

n = a i + b j +

⎤ ⎡ ⎤

x
y
z

x
y
z

⎣ ⎦ = ⎣ ⎦

R − φ,y R θ,z R α,x R − θ,z R φ,y

which aligns the axis of rotation with the x -axis, performs the rotation of P through
an angle α about the x -axis, and returns the axis of rotation back to its original
position. Therefore,

⎡ ⎤ ⎡ ⎤

cos φ 0 n φ cos θ sin θ 0

⎣ ⎦ , ⎣ ⎦

R φ,y = 0 1 0 R − θ,z = − sin θ cos θ 0

− sin φ 0 cos φ 0 0 1

⎡ ⎤ ⎡ ⎤

10 0 cos θ − sin θ 0

⎣ ⎦ , ⎣ ⎦

R α,x = 0 cos α − sin α R θ,z = sin θ cos θ 0

0 n α cos α 0 0 1

⎡ ⎤

cos φ sin φ

01 0
sin φ

0 −

⎣ ⎦ .

R − φ,y =

0 cos φ

Let

⎡ ⎤

a 11 a 12 a 13

⎣ ⎦

R − φ,y R θ,z R α,x R − θ,z R φ,y = a 21 a 22 a 23

a 31 a 32 a 33

where by multiplying the matrices together we find that:

cos 2 φ cos 2 θ cos 2 φ sin 2 θ cos α sin 2 φ cos α a 11 = + +

a 12 = cos φ cos θ sin θ − cos φ sin θ cos θ cos α − sin φ cos θ sin α

cos φ sin φ cos 2 θ cos φ sin φ sin 2 θ cos α sin 2 φ sin θ sin α a 13 = + +

cos 2 φ sin θ sin α

+ −

cos φ sin φ cos α

a 21 = sin θ cos θ cos φ − cos θ sin θ cos φ cos α + cos θ sin φ sin α

sin 2 θ + cos 2 θ cos α

a 22 =

a 23 = sin θ cos θ sin φ − cos θ sin θ sin φ cos α − cos θ cos φ sin α

cos φ sin φ cos 2 θ + cos φ sin φ sin 2 θ cos α − cos 2 φ sin θ sin α a 31 =

− cos φ sin φ cos α

a 32 = sin φ cos θ sin θ − sin φ sin θ cos θ cos α + cos φ cos θ sin α

sin 2 φ cos 2 θ sin 2 φ sin 2 θ cos α

a 33 = + − cos φ sin φ sin θ sin α

cos 2 φ cos α. + cos φ sin φ sin θ sin α +