Transformations in Three Dimensions - Computer Graphics: Principles and Practice - page 266

will be sent to infinity exactly if, after transformation, its w -coordinate is zero, that
is, if

Ax + By + Cz + D = 0. (11.7)

That equation defines a plane in 3-space.

Inline Exercise 11.2: In the case described above in which a projective trans-
formation is actually affine, which points in xyzw -coordinates form the “plane
sent to infinity”? It's important to include w in your computation.

11.2 Rotations

Rotations in 3-space are much more complicated than those in the plane, but much
of that complexity is of little significance for the casual user. We therefore present
the essentials in this section, but we provide a much longer discussion of rotations
in the web materials for this chapter.
We begin with some easily derived formulas that you're likely to use often.
Then we'll discuss how to use notions like pitch, roll, and yaw (which are called
Euler angles) to describe rotations, and how to describe a rotation by giving an
axis of rotation and an angle through which to rotate (Rodrigues' formula), as
well as how to find the axis and angle for a rotation (a computation that's also
due to Euler). Both of these descriptions of rotations have limitations that make
them unsuitable for interpolating between rotations, so we'll consider a third way
to describe rotations: To each point q of the sphere S 3 in four-dimensional space
R 4 , we can associate a rotation K ( q ) in a very natural way. There's a small prob-
lem, however: The points q and

q of S 3 correspond to the same rotation, so our
correspondence is two-to-one. It turns out to still be easy to use this description of
rotations to perform interpolation.

−

11.2.1 Analogies between Two and Three Dimensions
Rotations in two dimensions given by matrices of the form
cos

θ − sin θ

(11.8)

sin θ cos θ

generalize nicely to three dimensions and higher. For instance, we can take the
matrix for rotation through the angle θ in two dimensions and expand it to get

⎡ ⎤

cos θ − sin θ 0

⎣ ⎦ ,

R xy ( θ )= sin θ cos θ 0 (11.9)

0 0 1

which is the rotation by angle in the xy -plane of 3-space. As we mentioned

in Chapter 10, this is also sometimes called rotation about z by the angle

θ

. The
advantage of calling it rotation in the xy -plane is that there is an easy mnemonic
associated to it: For small values of

θ

, the unit vector in the x -direction is rotated
toward the unit vector in the y -direction. Corresponding statements are true of R yz

θ