Mechanics 2: Linear and Rotational Dynamics - 3D Math Primer for Graphics and Game Development - page 604

rotation as a quaternion. Although both functions express the same value,
they are different “data types.”
The analog of linear velocity and acceleration are called angular ve-
locity and angular acceleration. We denote them as ω(t) and α(t), re-
spectively, and both of these quantities are 3D vectors, or infinitesimal
exponential maps (see Section 8.4) if you will. We were able to define lin-
ear velocity as the time derivative of position, but things are a bit more
complicated with orientation. In general, ω(t) is not the derivative of the
orientation in any format (even exponential map). We have more to say
on this when we look at what the derivatives of angular values really are
in Section 12.6.4.
So our first order of business is to understand how to express and mea-
sure angular velocity. This is tricky, not just because rotation in 3D is more
complicated than position, but also because there are two slightly different
types of angular velocity. The first is known as spin angular velocity, and
the second is orbital angular velocity. Spin angular velocity describes the
rate of change of orientation of an object and is not affected by translation
of the object. Orbital angular velocity is actually not concerned with ori-
entation at all; instead, it measures the rate at which the position of an
object traces out an orbit around some other point. We already introduced
orbital angular velocity in Section 11.8.2, so if you skipped that section,
now would be a good time to review it.
To see the relationship between spin angular velocity and orbital angular
velocity, let's look at an example. Consider an object that is rotating about
its center of mass c , which is fixed in space. To describe this rotation, we
must specify two things. First, we describe the direction of the rotation; we
choose to do this by naming n , a unit vector parallel to the axis of rotation
whose sign (in combination with the left hand rule) establishes a direction
of positive rotation. Note that n tells us only the direction of the axis;
the position comes from our assumption that the axis passes through the
center of mass c . The other element necessary to describe the rotation is,
of course, the rate of rotation, which we measure in radians per unit time
and denote by the scalar ω. Now, we can define the spin angular velocity of
the rigid body by the angular velocity vector ω (note the boldface), which
is simply the multiplication of the rotation rate with the axis:
ω = ω n .
These ideas should be familiar to you, if you read Section 8.4, which talked
about exponential maps, and Section 11.8.2, which discussed uniform cir-
cular velocity of a particle and defined orbital velocity. If so, then you
can probably already see the connection between spin angular velocity and
orbital angular velocity.