Graphics Reference
In-Depth Information
and
R
zx
, which are written below. There's another advantage: While rotations in
3-space always have an axis (see the web material for a proof) those in 2-space
do not (e.g., there's no vector in
R
2
left invariant by rotation through 30
◦
), and
neither do those in 4-space. But in all cases rotations can be described in terms of
planes of rotation.
The analogous rotations in the
yz
- and
zx
-planes are given by
⎡
⎣
⎤
⎦
and
1
0
0
0
cos
θ −
sin
θ
R
yz
(
θ
)=
(11.10)
0
sin
θ
cos
θ
⎡
⎤
cos
θ
θ
01 0
0
sin
⎣
⎦
,
R
zx
(
θ
)=
(11.11)
−
sin
θ
0
cos
θ
which can also be called rotation about
x
and rotation about
y
, respectively.
In contrast to the two-dimensional situation, where we found that the set of
3
×
3 rotation matrices was one-dimensional, in three dimensions the set
SO
(
3
)
of 3
3 rotation matrices is
three
-dimensional. It is not, however, just a three-
dimensional Euclidean space. One way to see it's three-dimensional is to find a
mostly one-to-one mapping from an easy-to-understand three-dimensional object
to
SO
(
3
)
(just as the latitude-longitude parameterization of the 2-sphere shows
us that the 2-sphere is two-dimensional). We'll actually describe three such map-
pings, each with its own advantages and disadvantages. The first of these map-
pings is through
Euler angles.
This mapping is “mostly one-to-one,” in much the
same way that the mapping of latitude and longitude to points on the globe is
mostly one-to-one: Each point on the international date line has two longitudes
(180E and 180W), and each pole has infinitely many longitudes, but each other
sphere point corresponds to a single latitude-longitude pair.
×
Euler angles
are a mechanism for creating a rotation through a sequence of three
simpler rotations (called roll, pitch, and yaw). This decomposition into three sim-
pler rotations can be done in several ways (yaw first, roll first, etc.); unfortunately,
just about every possible way is used in
some
discipline. You'll need to get used
to the idea that there's no single correct definition of Euler angles.
The most commonly used definition in graphics describes a rotation by Euler
angles
(
)
as a product of three rotations. The matrix
M
for the rotation is
therefore a product of three others:
φ
,
θ
,
ψ
M
=
R
yz
(
ψ
)
R
zx
(
θ
)
R
xy
(
φ
)
.
(11.12)
Thus, objects are first rotated by angle
φ
in the
xy
-plane, then by angle
θ
in the
zx
-plane, and then by angle
ψ
in the
yz
-plane. The number
φ
is called pitch,
θ
is
ψ
called yaw, and
is called roll. If you imagine yourself flying in an airplane (see
Figure 11.1) along the
x
-axis (with the
y
-axis pointing upward) there are three
direction changes you can make: Turning left or right is called
yawing,
pointing
up or down is called
pitching,
and rotating about the direction of travel is called
rolling.
These three are independent in the sense that you can apply any one with-
out the others. You can, of course, also apply them in sequence.
