Rotation in Three Dimensions - 3D Math Primer for Graphics and Game Development - page 287

y = ( m31 − m13 ) ∗ m u l t ;

z = ( m12 − m21 ) ∗ m u l t ;

br e ak ;

c a s e 1 :

x = b i g g e s t V a l ;

w = ( m23 − m32 ) ∗ m u l t ;

y = ( m12 + m21 ) ∗ m u l t ;

z = ( m31 + m13 ) ∗ m u l t ;

br e ak ;

c a s e 2 :

y = b i g g e s t V a l ;

w = ( m31 − m13 ) ∗ m u l t ;

∗

x = ( m12 + m21 ) m u l t ;

∗

z = ( m23 + m32 ) m u l t ;

br e ak ;

c a s e 3 :

z = b i g g e s t V a l ;

w = ( m12 − m21 ) ∗ m u l t ;

x = ( m31 + m13 ) ∗ m u l t ;

y = ( m23 + m32 ) ∗ m u l t ;

br e ak ;

}
Listing 8.5
Converting a rotation matrix to a quaternion

8.7.5 Converting Euler Angles to a Quaternion
To convert an angular displacement from Euler angle form to quaternion, we
use a technique similar to the one used in Section 8.7.1 to generate a rotation
matrix from Euler angles. We first convert the three rotations to quaternion
format individually, which is a trivial operation. Then we concatenate these
three quaternions in the proper order. Just as with matrices, there are
two cases to consider: one when we wish to generate an object-to-upright
quaternion, and a second when we want the upright-to-object quaternion.
Since the two are conjugates of each other, we walk through the derivation
only for the object-to-upright quaternion.
As we did in Section 8.7.1, we assign the Euler angles to the variables h,
p, and b. Let h , p , and b be quaternions that perform the rotations about
the y, x, and z-axes, respectively:

2
4 cos(h/2)

3
5

2
4 cos(p/2)

3
5

2
4 cos(b/2)

3
5

0 1 0 1 0 1

0
sin(h/2)
0

sin(p/2)
0
0

0
0
sin(b/2)

h = , p = , b = .

@ A @ A @ A

Now we concatenate these in the correct order. We have two sources of
“backwardness,” which cancel each other out. We are using fixed-axis rota-
tions, so the order of rotations actually is bank, pitch, and lastly heading.