Quaternions - Rotation Transforms for Computer Graphics - page 64

Next, we compute q 2 q 1
q 1 = s 1 − v 1

q 2 = s 2 − v 2

q 2 q 1 = (s 2 − v 2 )(s 1 − v 1 )

= s 1 s 2 − v 1 · v 2 + s 2 ( − v 1 ) + s 1 ( − v 2 ) + ( − v 2 ) × ( − v 1 )

= s 1 s 2 − v 1 · v 2 − s 1 v 2 − s 2 v 1 − v 1 × v 2 (5.4)

and as ( 5.3 ) equals ( 5.4 ), ( q 1 q 2 ) ∗ = q 2 q 1 .

5.9 The Inverse Quaternion

Given a quaternion q we can compute its inverse q − 1 as follows.

By definition, we require that
qq − 1 q − 1 q = = 1 . (5.5)

First, we multiply ( 5.5 )by q ∗

q ∗ qq − 1 q ∗ q − 1 q q ∗

= = (5.6)

and from ( 5.6 ) we can write

q ∗
q ∗ q =

q ∗

q − 1 =

2 .

| q |

If q is a unit quaternion, then q − 1 q ∗ , which is useful when reversing a rotational =

sequence. Therefore, as

( q 1 q 2 ) ∗ = q 2 q 1

then

( q 1 q 2 ) − 1 q − 1

2

q − 1
1

= .

For completeness let's evaluate the inverse of q where

1
√ 3

1
√ 3

1
√ 3

q = 1 + i + j + k

1
√ 3 i

1
√ 3 j

1
√ 3 k q ∗ = 1 − − −

1
3 +

1
3 +

1
3 =

2

| q | = 1 + 2

1

√ 3 k

1
2

1
√ 3 i

1
√ 3 j

1

q − 1 = − − −

1
2 −

1
√ 12

1
√ 12

1
√ 12 = i − j − k .