Graphics Reference
In-Depth Information
2.14 Change of axial system
One useful application for direction cosines is in the transformation of points from one
coordinate system to another. For example, consider the problem of calculating the coordinates
of a 2D point relative to a system of axes that have been rotated relative to the original axes,
as shown in Fig. 2.43.
Y
Y
′
X
′
θ
φ
90
°
+
θ
θ
X
Figure 2.43.
X
is rotated relative to X and has direction cosines cos and cos .
But as
sin
=
cos
X
has direction cosines cos and sin.
Similarly, Y
is rotated relative to Y and has direction cosines cos 90
+
and cos .
But as
cos 90
+
=−
sin
Y
has direction cosines
sin and cos .
If the axial system is rotated , this is equivalent to rotating a point in the opposite direction
−
−
.
The rotation transform for an angle is
cos
−
sin
sin
cos
Therefore, the transform for an angle
is
cos
−
−
−
sin
−
sin
−
cos
−
which becomes
cos
sin
−
sin
cos
