Game Development Reference
In-Depth Information
(those for which
e
< 2π) and form a one-to-one correspondence with the
rotation matrices. This is the essence of Euler's rotation theorem.
Now let's consider concatenating multiple rotations. Let's say
e
1
and
e
2
are two rotations in exponential map format. The result of performing
the rotations in sequence, for example,
e
1
and then
e
2
, is not the same as
performing a single rotation
e
1
+
e
2
. We know this cannot be true, because
ordinary vector addition is commutative, but three-space rotations are not.
Assume that
e
1
= [90
o
,0,0], and
e
2
= [0,90
o
,0]. With our conventions, this
is a 90
o
downward pitch rotation, and a 90
o
heading rotation to the east.
Performing
e
1
followed by
e
2
, we would end up looking downward with our
head pointing east, but doing them in the opposite order, we end up “on our
ear” facing east. But what if the angles were much smaller, say 2
o
instead of
90
o
? Now the ending rotations are more similar. As we take the magnitude
of the rotation angles down, the importance of the order decreases, and at
the extreme, for “infinitesimal” rotations, the order is completely irrelevant.
In other words, for infinitesimal rotations, exponential maps can be added
vectorially. Infinitesimals are important topics from calculus, and they
are at the heart of defining rate of change. We look at these topics in
Chapter 11, but for now, the basic idea is that exponential maps do not
add vectorially when used to define an amount of rotation (an angular
displacement or an orientation), but they do properly add vectorially when
they describe a rate of rotation. This is why exponential maps are perfectly
suited for describing angular velocity.
Before we leave this topic, a regretful word of warning regarding ter-
minology. Alternative names for these two simple concepts abound. We
have tried to choose the most standard names possible, but it was di
cult
to find strong consensus. Some authors use the term “axis-angle” to de-
scribe both of these (closely related) methods and don't really distinguish
between them. Even more confusing is the use of the term “Euler axis”
to refer to either form (but not to Euler angles!). “Rotation vector” is an-
other term you might see attached to what we are calling exponential map.
Finally, the term “exponential map,” in the broader context of Lie algebra,
from whence the term originates, actually refers to an operation (a “map”)
rather than a quantity. We apologize for the confusion, but it's not our
fault.
8.5
Quaternions
The term quaternion is somewhat of a buzzword in 3D math. Quaternions
carry a certain mystique—which is a euphemismistic way of saying that
many people find quaternions complicated and confusing. We think the way
quaternions are presented in most texts contributes to their confusion, and
Search WWH ::
Custom Search