Rotating Vectors - Vector Analysis for Computer Graphics - page 193

where

s 2 x 2 y 2 z 2

+ + + = 1

and the pure quaternion v, which contains the position vector v :

= +

v 0 v

Therefore,

qv ¯ q = s + q 0 + v s − q

Multiplying the first two quaternions gives

¯ = − · + + × −

qv q q v s v q v s q

Multiplying these quaternions gives

qv q ¯ =− q · v s − s v + q × v · − q + − q · v − q + s s v + q × v + s v + q × v × − q

Rearranging and simplifying terms gives

s 2 v

qv ¯ q =− s q · v + s v · q + − q · v − q + + s q × v + s v × − q + q × v × − q

qv q ¯ = q · v q + s 2 v + s q × v + s v × − q + q × v × − q

Using the triple vector product identity gives

q × v × − q =− q · qv + v · q q

Therefore,

¯ = · +

s 2 v

+ × + × − · + ·

qv q q v q s q v s q v q qv v q q

= s 2

v

2

qv q ¯ − q + 2 q · v q + 2s q × v (7.24)

The next task is to convert Eq. (7.24) into a matrix, which we will do in three steps.

Converting s 2

v :

2

− q

2

x 2 y 2 z 2 s 2 q = + + = 1 −

Therefore,

− 1

s 2 =

2

s 2

− =

s 2

−

2s 2

−

q 1

and

s 2 v

= 2s 2 1 v = 2s 2 1 x v i + 2s 2 1 y v j + 2s 2 1 z v k

2

− q − − − −

⎡ ⎤ ⎡ ⎤

2s 2 10 0

0 s 2

− x v

y v
z v

s 2 v

2

⎣ ⎦ ⎣ ⎦

− q = − 10

2s 2

0 0 − 1