Spatial acceleration

In physics the study of rigid body motion provides for several ways of defining the acceleration state of a rigid body. The classical definition of acceleration entails following a single particle/point along the rigid body and observing its changes of velocity. In this article the notion of spatial acceleration is explored, which entails looking at a fixed (unmoving) point in space and observing the changes of velocity of whatever particle/point happens to coincide with the observation point. This is similar to the acceleration definition fluid dynamics where typically one can measure velocity and/or accelerations on a fixed locate inside a testing apparatus.
Definition.
Consider a moving rigid body and the velocity of a particle/point "P" along the body being a function of the position and velocity of a center particle/point "C" and the angular velocity formula_1. 
The linear velocity vector formula_2 at "P" is expressed in terms of the velocity vector formula_3 at "C" as:
formula_4
where formula_1 is the angular velocity vector.
The material acceleration at "P" is:
<math>\vec a_P = \frac. Page 41 (link: Google Books ) defines spatial accelerations for use in rigid body mechanics.