ME/CS 133(a): Notes on Rotations

Consider a 3-dimensional rigid body, B. Let C denote the point in B which is fixed during subsequent spherical motions. Choose a fixed reference frame, F whose origin lies at point C. To understand what happens to a rigid body during a spherical motion, select an arbitrary point, P0, in B. Let ~v0 denote the vector from the origin of this fixed reference frame to P0. I.e., ~v0 denotes the coordinates of P0 as seen by an observer in frame F . Let the body undergo a spherical displacement. After the displacement, let the new location of the point P0 be denoted by P1. Similarly, let ~v1 denote the coordinates of P1, as seen in frame F . Let us assume that the transformation of the points in B during the spherical displacement can be represented by the action of a 3 × 3 matrix, A, acting on the vector of particle coordinates: ~v1 = A~v0 . Because the body is rigid, the distance between the fixed point of the motion (which is also the origin of the fixed reference frame) and the given particle is constant. That is: