ABSTRACT
Spatial rigid-body rotations are defined as motions that preserve the distance between points in a body before and after the motion and leave one point fixed under the motion. By definition a motion must be physically realizable, and so reflections are not allowed. If X1 and X2 are any two points in a body before a rigid motion, then x1, and x2 are the corresponding points after rotation, and
d(x1, x2) = d(X1, X2) where
d(x, y) = ||x − y|| = √
(x1 − y1)2 + (x2 − y2)2 + (x3 − y3)2
is the Euclidean distance. We view the transformation from Xi to xi as a function x(X, t). By appropriately choosing our frame of reference in space, it is possible to make the pivot point (the
point which does not move under rotation) the origin. Therefore, x(0, t) = 0. With this choice, it can be shown that a necessary condition for a motion to be a rotation is
x(X, t) = A(t)X where A(t) ∈ IR3×3 is a time-dependent matrix.
