On the factorization of rotations with examples in diffractometry

Abstract

An analysis of compound rotations, such as occur in eulerian cradles, is presented in terms of a calculus of rotation axes, without reference to the associated coordinate transformations. The general case of three rotation shafts mounted on one another, with any relation between them at datum zero, is presented. The problem and its solution may be represented entirely in terms of a plane octagon in which four sides have directions that are instrumental constants and the other four sides have lengths that are instrumental constants. When the first four sides are given lengths that express both the rotation angle and the axial direction of the required rotation, then the remaining four sides have directions that directly express the rotations in the drive shafts, that will generate the required rotation. Analytic expressions are given for the shaft setting angles in the general case. If the first and third axes are parallel and the intermediate one perpendicular to these at datum zero (as in the four-circle diffractometer) then these reduce to θ1 = arctan (μ, σ) + [arctan (λ, v) - ψ -½8π], θ2 = 2s arcsin (λ2 + v2)½, θ3 = (μ, σ) - [arctan (λ, v) - ψ - ½8π], s = ± 1, 0 ≤ arcsin (λ2 + v2)½ ≤ ½π, in which λ, μ, v and σ are the four components of a rotation vector constructed such that λ, μ and v are the direction cosines of the rotation axis multiplied by sin½θ for a rotation angle θ and σ is cos½θ. ψ is a constant determined by the choice of directions to which λ and v are measured. The results for the general case are also expressed in terms of more conventional variables.