Vector Representation - Vector Analysis for Computer Graphics - page 56

A similar argument can be used for calculating the new coordinates of a point relative to a
rotated set of 3D axes. Once again, the point is rotated in the opposite direction to the axis
rotation. Although this can require three individual rotations about the x-, y-, and z-axes, the
final result can be expressed as a transform:
⎡ ⎤

r 11 r 12 r 13

⎣ ⎦

r 21 r 22 r 23

r 31 r 32 r 33

where

r 11 r 12 r 13 are the direction cosines of the rotated X -axis relative to the original X-, Y-, and
Z-axes, respectively,
r 21 r 22 r 23 are the direction cosines of the rotated Y -axis relative to the original X-, Y-, and
Z-axes, respectively,
r 31 r 32 r 33 are the direction cosines of the rotated Z -axis relative to the original X-, Y-, and
Z-axes, respectively.

Thus, the new coordinates of a point P x P y P z P are given by
⎡

⎤ ⎡ ⎤ ⎡ ⎤

x P
y P
z P

r 11 r 12 r 13 x P

y P
z P

⎣ ⎦ = ⎣ ⎦ · ⎣ ⎦

r 21 r 22 r 23

r 31 r 32 r 33

Note that

r 11 + r 12 + r 13 = 1

r 21 + r 22 + r 23 = 1

r 31 + r 32 + r 33 =

1

To demonstrate the above, consider the case of calculating the new coordinates of P 111,
where the axial system is rotated as illustrated in Fig. 2.45.

Y

X ′

P (1,1,1)

Y ′

Z ′

X

Z

Figure 2.45.