Rotation in Three Dimensions - 3D Math Primer for Graphics and Game Development - page 277

body axes, we do fixed-axis rotations, where the rotations are about the
upright axes. This means we actually do the rotations in the reverse order:
first bank, then pitch, and finally heading. See Section 8.3.2 if you don't
remember what fixed-axis rotations are.
In summary, the generation of the object-to-upright rotation matrix is
a straightforward concatenation of three simple rotation matrices,

M object→upright = BPH ,

where B , P , and H are the rotation matrices for bank, pitch, and heading,
which rotate about the z-, x-, and y-axes, respectively. We learned how to
compute these elementary rotation matrices in Section 5.1.2.

2
4

3
5

cosb sinb 0
− sinb cosb 0
0

Elementary rotation
matrices for bank, pitch,
and heading

B = R z (b) = ,

0 1

2 3

1 0 0

4 5

P = R x (p) = 0 cosp sinp ,

0 − sinp cosp

2 3

cosh 0 − sinh

4 5

H = R y (h) = 0 1 0 .

sinh 0 cosh

Putting it all together (and leaving out the messy math to actually do
the matrix multiplications), we have

M object→upright = BPH

Object-to-upright
rotation matrix from
Euler angles

2 3

chcb + shspsb sbcp −shcb + chspsb
−chsb + shspcb cbcp sbsh + chspcb
shcp

, (8.14)

4 5

=

−sp chcp

where we have introduced the shorthand notation

ch = cosh, cp = cosp, cb = cosb,

sh = sinh, sp = sinp, sb = sinb.

To rotate vectors from upright space to object space, we will use the
inverse of this object-to-upright matrix. We know that since a rotation
matrix is orthogonal, the inverse is simply the transpose. However, let's
verify this.
To visualize the upright-to-object transform, we imagine the undoing
the fixed-axis rotations. We first undo the heading, and then the pitch,