Quaternion Transforms in Space - Rotation Transforms for Computer Graphics - page 177

11.9 Converting a Rotation Matrix to a Quaternion

The matrix transform equivalent to qpq − 1 is

⎡ ⎤ ⎡ ⎤

2 (s 2 x 2 ) + − 1 2 (xy − sz) 2 (xz + sy) x u

y u
z u

qpq − 1

⎣ ⎦ ⎣ ⎦

2 (s 2 y 2 ) = 2 (xy + sz) + − 1 2 (yz − sx)

2 (s 2 + z 2 ) −

2 (xz − sy) 2 (yz + sx) 1

⎡ ⎤ ⎡ ⎤

a 11 a 12 a 13 x u

y u
z u

⎣ ⎦ ⎣ ⎦ .

= a 21 a 22 a 23

a 31 a 32 a 33

Inspection of the matrix shows that by combining various elements we can isolate
the terms of a quaternion s,x,y,z . For example, by adding the terms a 11 +

a 22 + a 33

we obtain:

a 33 = 2 s 2 x 2 − 1 + 2 s 2 y 2 − 1 + 2 s 2 z 2 − 1 a 11 + a 22 + + + +

2 x 2 z 2 − 6 s 2 y 2

= + + + 3

4 s 2 = − 1

therefore,

2 1

1

s =± + a 11 + a 22 + a 33 .

To isolate x , y and z we use

1
4 s (a 32 −

x = a 23 )

1
4 s (a 13 −

y = a 31 )

1
4 s (a 21 −

z = a 12 ).

We can confirm their accuracy using the matrix ( 11.9 ):

1 √ 2

√ 3

1
2

2
3 +

2
3 +

1
3 =

s =± +

√ 3
4 √ 2

2
3 +

2
3

1
√ 6 x = =

√ 3
4 √ 2

2
3 +

2
3

1
√ 6 y = =

√ 3
4 √ 2

1
3 −

1
3

z = = 0

which agree with the original values.
Say, for example, the value of s had been close to zero, this could have made the
values of x , y , z unreliable. Consequently, other combinations are available: