The dual Euler-Rodrigues formula in various mathematical forms and their intrinsic relations

This paper examines the variations and derivations of the dual Euler-Rodrigues formula from various mathematical forms, including the matrix in 6 × 6, the dual matrix, Lie group SE(3) of the exponential map of the Lie algebra se(3), and the dual quaternion conjugation, and investigates their intrinsic connections. Based on the dual Euler-Rodrigues formula, the axis, the dual rotation angle, and the new traces are obtained by using the properties of the skew-symmetric matrices. In decomposing the Chasles' motion, this paper examines two ways of realization of the motion based on the Mozzi-Chasles' axis. With the equivalent motion, the paper relates the finite displacement screw matrix, the exponential map, and the dual quaternion conjugation to the dual Euler-Rodrigues formula and reveals their connection with the Mozzi-Chasles axis screw, whose parameters are used to construct the Lie algebra, the dual Euler-Rodrigues formula, and the dual quaternion. Further, using the Mozzi-Chasles axis screw, the paper presents a complete geometrical interpretation, including both the translation and rotation, and associates it with the algebraic presentation. By decomposing the equivalent translation induced by the rotation, the paper presents the mapping between the compound translation and the secondary part of the Mozzi-Chasles axis screw. With this map and the compound translation, the paper hence reveals the intrinsic connection between various presentations of rigid body transformations by formulating them into the dual Euler-Rodrigues formula and presents the relations of the exponential map of the Mozzi-Chasles axis screw to the finite displacement screw matrix and the dual Euler-Rodrigues formula, leading to the understanding of the various forms of a rigid body displacement in correspondence to the dual Euler-Rodrigues formula.