Matrices and Linear Transformations - 3D Math Primer for Graphics and Game Development - page 143

Let's summarize the vectors we have computed:

v = ( v n ) n ,
v ⊥ = v − v
= v − ( v n ) n ,
w = n × v ⊥
= n ×

v − v

= n × v − n × v
= n × v − 0
= n × v ,

′ ⊥ = cosθ v ⊥ + sinθ w
= cosθ ( v − ( v n ), n ) + sinθ ( n × v ).

v

′

Substituting for v , we have

′ ′

v = v

⊥ + v
= cosθ ( v − ( v n ) n ) + sinθ ( n × v ) + ( v n ) n .

(5.1)

Equation (5.1) allows us to rotate any arbitrary vector about any arbitrary
axis. We could perform arbitrary rotation transformations armed only with
this equation, so in a sense we are done—the remaining arithmetic is es-
sentially a notational change that expresses Equation (5.1) as a matrix
multiplication.
Now that we have expressed v in terms of v , n , and θ, we can compute

what the basis vectors are after transformation and construct our matrix.
We're just presenting the results here; a reader interested in following each
step can check out Exercise 2.24:

′

2 3

5 T

n x 2 (1 − cosθ) + cosθ
n x n y (1 − cosθ) + n z sinθ
n x n z (1 − cosθ) − n y sinθ

′

4

p = 1 0 0 , p = ,

2 3

5 T

n x n y (1 − cosθ) − n z sinθ
n y 2 (1 − cosθ) + cosθ
n y n z (1 − cosθ) + n x sinθ

4

′

q = 0 1 0 , q = ,

2 3

5 T

n x n z (1 − cosθ) + n y sinθ
n y n z (1 − cosθ) − n x sinθ
n z 2 (1 − cosθ) + cosθ

4

′

r = 0 0 1 , r = .