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

Abstract

The wellknown 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 selfcontained, 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 خ±, خ², خ¸ 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 خ¸ 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 خ spanned by a and b, and one perpendicular to خ . Moreover, the vector in خ is always between a and b. (c) The rotation axes for the two possible composite rotations are symmetric with respect to خ . (d) The angle خ¸ decreases from خ±â€‰+ خ² to خ±â€‰âˆ’ خ² as the angle between a and b increases from 0 to د€. Finally, the Rodrigues equations are used to derive the equations corresponding to the composition of infinitesimal rotations, which are also derived using alternative approaches.