Game Development Reference

In-Depth Information

as affected by the previous rotation before they can be correctly applied. In other words,

you have to rotate
R
2
by
R
1
to get a new
R
2
before applying it. All this happens to work

out in such a way that you reverse the order of multiplication of rotation matrices when

they are defined in a rotating coordinate system.

Figure 11-1
shows a right-handed coordinate system that illustrates the directions of

positive rotation about each coordinate axis.

Figure 11-1. Right-handed coordinate system

Let's consider rotation around the z-axis where the point shown in
Figure 11-2
is rotated

through an angle θ.

Figure 11-2. Rotation around the z-axis

The coordinates of the point before the rotation are (
x
,
y
,
z
) and after the rotation the

coordinates are (
x
r
,
y
r
,
z
r
). The rotated coordinates are related to the original coordinates

and the rotation angle by the following:

x
r
= x cos θ − y sin θ

y
r
= x sin θ + y cos θ

z
r
= z