Transformations - Transformations and Projections in Computer Graphics - page 40

Okay, so I assume going into this tutorial that you know how to perform matrix
multiplication. I don't care to explain it, and it's available all over the Internet.
However, once you know how to perform that operation, you should be good to go for
this tutorial.

(Found on the Internet)

It is therefore easy to identify the axis of rotation for each of the three rotation
matrices of Equation (1.29), but what about their direction of rotation? To figure out
the directions, we select θ =90 ◦ and substitute sin θ = 1 and cos θ =0. Simpletestsina
right-handed coordinate system show that the first matrix of Equation (1.29) (rotation
about the z axis) rotates point (1 , 0 , 0) to (0 , 1 , 0) and point (0 , 1 , 0) to (1 , 0 , 0). Thus,

when we observe this 90 ◦ rotation looking in the direction of positive z , the rotation
is counterclockwise (Figure 1.23a). The second matrix, however, behaves differently.
It rotates point (1 , 0 , 0) to (0 , 0 , − 1) and point (0 , 0 , 1) to (1 , 0 , 0). When we observe
this 90 ◦ rotation about the y axis looking in the direction of positive y , the rotation is
clockwise (Figure 1.23b). The third matrix (rotation about the x axis) rotates point
(0 , 1 , 0) to (0 , 0 ,

−

1) and point (0 , 0 , 1) to (0 , 1 , 0). When we observe this 90 ◦ rotation
looking in the direction of positive x , the rotation is counterclockwise (Figure 1.23c).
We therefore decide (somewhat arbitrarily) to switch the signs (positive and neg-
ative) of the sine functions in the matrices that rotate about the z and x axes. The
result,
⎛
⎝

−

⎞
⎠

⎛
⎝

⎞
⎠

⎛
⎝

⎞
⎠

cos θ sin θ 00 cos θ 0 − sin θ 0 1 0 0 0

− sin θ cos θ 00 0 1 0 0 0 θ sin θ 0

, , ,

0 0 1 0 sin θ 0 θ 0 0 − sin θ cos θ 0

0 0 0 1 0 0 0 1 0 0 0 1

(1.30)
is a set of three rotation matrices that rotate a point about the three coordinate axes in
such a way that if we look in the positive direction of that axis, the rotation is clockwise.
(Surprisingly, it turns out that there is an elegant way to specify the direction
of rotation that's generated by the rotation matrices of Equation (1.29), and this is
described below.)
The rotation matrices of Equations (1.29) and (1.30) are simple but not very useful
because in practice we rarely know how to break a general rotation into three rotations
about the coordinate axes. There are some cases, however, where rotations about the
coordinate axes are common. One such case is discussed in Section 2.2; two more are
presented here.
Case 1: Rotations about the coordinate axes are common in the motion of a sub-
marine or an airplane. These vehicles have three degrees of freedom and have three
natural, mutually perpendicular axes of rotation that are called roll , pitch ,and yaw
(Figure 1.24). Roll is a rotation about the direction of motion of the vehicle. An air-
plane rolls when it banks by dipping one wing and lifting the other. Pitch is an up or
down rotation about an axis that goes through the wings. An airplane uses its elevators
for this. Yaw is a left-right rotation about a vertical axis, accomplished by the rudder.
These terms originated with sailors because a ship can yaw and also has limited roll
and pitch capabilities.