Rigid-body motion

Now suppose that a rigid body is moving through ${\mathbb{R}}^3$. In this case, all its points move together. How can we easily describe this motion? Recall from Section 3.2 that translations or rotations may be applied. First, consider a simple case. Suppose that rotations are prohibited, and the body is only allowed to translate through space. In this limited setting, knowing the position over time for one point on the body is sufficient for easily determining the positions of all points on the body over time. If one point has changed its position by some $(x_t,y_t,z_t)$, then all points have changed by the same amount. More importantly, the velocity and acceleration of every point would be identical.

Once rotation is allowed, this simple behavior breaks. As a body rotates, the points no longer maintain the same velocities and accelerations. This becomes crucial to understanding VR sickness in Section 8.4 and how tracking methods estimate positions and orientations from sensors embedded in the world, which will be discussed in Chapter 9.