Page 120 - Computer Graphics Handout

We can find the matrices for rotation about the individual axes directly from the results of the two-dimensional rotation that we
developed in Section 3.7.2. We saw that the two-dimensional rotation was actually a rotation in three dimensions about the z-axis
and that the points remained in planes of constant z. Thus, in three dimensions, the equations for rotation about the z-axis by an
angle θ are
x_ = x cos θ − y sin θ ,
y_ = x sin θ + y cos θ ,
z_ = z ,
or, in matrix form,
,
p = Rzp,
where

We can derive the matrices for rotation about the x- and y-axes through an identical argument. If we rotate about the x-axis, then
the x values are unchanged, and we have a two-dimensional rotation in which points rotate in planes of constant x; for rotation

about the y-axis, the y values are unchanged. The matrices are

The signs of the sine terms are consistent with our definition of a positive rotation in a right-handed system.
Suppose that we let R denote any of our three rotation matrices. A rotation by θ can always be undone by a subsequent rotation by
−θ; hence,
R−1(θ) = R(−θ).
In addition, noting that all the cosine terms are on the diagonal and the sine terms are off-diagonal, we can use the trigonometric
identities
cos(−θ) = cos θ
sin(−θ)=−sin θ
to find
R−1(θ) = RT(θ).
In Section 3.10.1, we show how to construct any desired rotation matrix, with a fixed point at the origin, as a product of individual
rotations about the three axes
R = RzRyRx .
Using the fact that the transpose of a product is the product of the transposes in the reverse order, we see that for any rotation
matrix,
T
R = R .
−1
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