The Rodrigues Equations for the Composition of Finite Rotations: A Simple Ab Initio Derivation and Some Consequences

The well-known Rodrigues' equations for the composition of two finite rotations were introduced in 1840 without a detailed derivation. Although later derivations are available, the one presented here is very simple, is based on rotation results due to Euler, is essentially self-contained, and makes use of analysis techniques familiar to engineering students. As the Rodrigues equations have a strong connection to quaternions, this matter is considered in some detail. Then the equations are used to derive some general results concerning the angle and axis of the composite rotation. If alpha, beta, theta and a, b, d indicate the rotation angles and axes for the two individual rotations and for the composite rotation, respectively, some of the results are as follows. (a) The value of theta does not depend on the order of the two rotations. (b) The vector d is equal to the sum of two vectors, one in the plane Pi spanned by a and b, and one perpendicular to Pi. Moreover, the vector in Pi is always between a and b. (c) The rotation axes for the two possible composite rotations are symmetric with respect to Pi. (d) The angle h decreases from alpha + beta to alpha - beta as the angle between a and b increases from 0 to pi. Finally, the Rodrigues equations are used to derive the equations corresponding to the composition of infinitesimal rotations, which are also derived using alternative approaches.