Page 116 - Computer Graphics Handout

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We expand this form to three dimensions in Section 3.9. Note three features of this transformation that extend to other rotations:
1. There is one point—the origin, in this case—that is unchanged by the rotation. We call this point the fixed point of the
transformation. Figure 3.35 shows a two-dimensional rotation about a fixed point in the center of the object
rather than about the origin of the frame.
2. Knowing that the two-dimensional plane is part of three-dimensional space, we can reinterpret this rotation in three dimensions.
In a right-handed system, when we draw the x- and y-axes in the standard way, the positive z-axis comes out of the page. Our
definition of a positive direction of rotation is counterclockwise when we look down the positive z-axis toward the origin. We use
this definition to define positive rotations about other axes.
3. Rotation in the two-dimensional plane z = 0 is equivalent to a threedimensional rotation about the z-axis. Points in planes of
constant z all rotate in a similar manner, leaving their z values unchanged.
We can use these observations to define a general three-dimensional rotation that is independent of the frame.We must specify
the three entities shown in Figure 3.36:
a fixed point (Pf ), a rotation angle (θ), and a line or vector about which to rotate. For a given fixed point, there are three degrees of
freedom: the two angles necessary to specify the orientation of the vector and the angle that specifies the amount of rotation about
the vector.

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