What is angular velocity in 3-dimensional space? (Revised) If I'm not mistaken, there are analogies between the translational dynamics of a rigid body and the rotational dynamics of that body. For example, the position of a rigid body is analogous to the orientation of that body. What property of a rigid body, from a rotational aspect, is analogous to the rigid body's velocity? (I would ask the similar question with respect to acceleration, but then this question would get disqualified by the moderators.)
In 2-dimensional space, I'm guessing that angular-velocity is analogous to velocity. But in 3-dimensional space, the way an object can be rotating can be much more complex. For example, the body may be rotating about an axis which itself is also rotating about some other axis. I imagine these axes all stemming from the center of mass of the rigid body.
Note that a single axis/angle pair is enough to describe a body's orientation at any given time, but it is not always sufficient to describe how that body is rotating in 3-dimensional space.
 A: First off, the planar (2D) and spatial (3D) cases are not two distinct cases. Rather there is only one case, the 3D, and the planar case is just a projection of the 3D cases where rotations are constrained along a single direction and translational motion along the plane perpendicular to it. So for 2D cases, in-plane motion is translation and out-of-plane motion is rotation. So looking at rigid body motions in 3D is the more general case, and once that is understood you can always project down to 2D.
So velocity is velocity (as a vector with direction and magnitude) in both 2D and 3D. The same with angular velocity (as a vector with direction and magnitude) for both 2D and 3D. For the planar case (translational) velocity has 2 in-plane components, and (rotational) velocity has 1 out-of-plane component. So even though $\omega$ looks like a scalar, it is really a vector of known direction, and only the magnitude is specified.
Now for the analogy between translational and rotational motion, I am not a big fan because it implies that they are two completely unrelated frameworks that kind of work the same way. But they are not. They are one and the same framework and what ties them together is geometry.
So here is why translational and rotational motion of a rigid body are one and the same thing almost. I can illustrate things better in 2D, but as I mentioned above, you can generalize to 3D if you allow rotations about arbitrary directions.
Imagine you are given a special magnifying glass, which you can use the measure the velocity of whatever particles are under it. Imagine you have a long body which is pivoting about a point B, and you are measuring the velocity at point A

Now the individual points might be moving (point A) or not (point B), but the rigid body as a whole is just rotating. The full motion of the body is specified by the location of point B and the rotational properties of the body. We can look at the rotational velocity $\omega$, and find the velocity of any point on the body with $$ v_A = \omega \, r_A $$ in 2D, and $$ \vec{v}_A = \vec{\omega} \times \vec{r}_A $$ in 3D.
In fact, if point B was also moving (part of a slot for example), then there would be some other point that would act as a pivot, even if instantaneously. From the pivot point all other motions of the body are derived.
The general motion of a rigid body on a plane is an instantaneous rotation about a point, call the instant center of rotation. There is an exception or special case to this when all the parts of the body have the same velocity and the whole rigid body is translating in space. This is commonly viewed as the body rotating about a point at infinity (really far away). Or you can consider it a second case where a body is either rotating about a point, or all points translate with the same velocity.
In 3D, the same applies, but you can also have a translational motion out-of-plane as well as the rotational motion in-plane at the same time. Think of the point B translating out-of-plane, forming a line in space. This line is the rotation axis of the body.
The general motion of a rigid body in 3D is an instantaneous rotation about a line, called the instance rotation axis, in addition to a parallel translation along this axis. Now the special case of translation in 2D is part of the general case, as you can have zero rotation and the above definition still holds true. There are no exceptions in 3D, as the general case is in fact general and all-encompassing.
The above is known as Chasle's Theorem and forms the basis of Screw Theory of rigid body dynamics.
A: Before getting to the question, let me first clarify a couple of points. Since the rotation group is three dimensional, it takes three angles to specify the orientation of a body in three dimensions with respect to some reference orientation. For example, we could specify the three Euler angles of the rotation.
Furthermore, at any given instant in time, the body will be rotating about a single axis, known as the axis of rotation. This axis may simply be time varying.
Now for angular velocity. I will first point out that this is not a scalar, as many introductory sources might lead one to believe. It is, in fact, a three dimensional vector which points along the axis of rotation and has magnitude equal to the radians per second spun by the body. Negating the angular velocity is equivalent to changing the direction of rotation about the axis.
Angular acceleration, since you mentioned it, is defined to be the time derivative of the angular velocity, understood in the way I described as a 3-vector.
A: If I understood, you are making a kind of analogy table, where position in translational dynamics is equivalent to angular orientation in rotational dynamics.
Following this line, what is the analogy pair of linear velocity?
There is a problem because velocity is conserved for translational movement without applied forces, but angular velocity is not. The rotating body can wobble.
But a good pair analogy is linear momentum, that is also conserved, and angular momentum. Even a wobbling rigid body has a constant angular momentum without applied torques.
