Rotation Transforms in Space - Rotation Transforms for Computer Graphics - page 137

To find a 13 :

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

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

b 2 1

b 2

a c a c

b 2 b 2 cos α

= √ 1 √ 1 − + √ 1 √ 1 + b sin α

b 2 b 2

− − − −

a c

− √ 1 √ 1 cos α

b 2 b 2

− −

b 2 ac

= ac + ac

b 2 ) cos α

+ b sin α −

b 2 ) cos α ( 1 − ( 1 −

= ac + ac (b 2

− 1 )

b 2 ) cos α + b sin α

( 1 −

= ac( 1 − cos α) + b sin α

a 13 = acK + b sin α.

Using similar algebraic methods, we discover that:

a 21 = abK + c sin α

b 2 K a 22 = + cos α

a 23 = bcK − a sin α

a 31 = acK − b sin α

a 32 = bcK + a sin α

c 2 K a 33 = + cos α

and our original matrix transform becomes:
⎡
⎣

⎤
⎦ =

⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦

x p
y p
z p

a 2 K +

cos α bK − c sin αacK + b sin α

x p
y p
z p

2 K

abK + c sin α + cos α cK − a sin α

2 K

acK − b sin α bcK + a sin α + cos α

where

K = 1 − cos α.

9.6.2 Vectors

Now let's solve the same problem using vectors. Figure 9.11 shows a view of the
geometry associated with the task at hand. For clarification, Fig. 9.12 shows a cross-
section and a plan view of the geometry.
The axis of rotation is given by the unit vector:

n ˆ = a i + b j + c k .