On the factorization of rotations with examples in diffractometry

R. Diamond
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引用次数: 4

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.
以衍射法为例讨论旋转的因式分解
复合旋转的分析,如发生在欧拉云台,提出了一个微积分的旋转轴,而不涉及相关的坐标变换。给出了三个相互安装的转轴的一般情况,它们之间在基准零点处有任何关系。这个问题和它的解可以完全用一个平面八边形来表示,其中四个边的方向是仪器常数,另外四个边的长度是仪器常数。当给定前四个边的长度表示所需旋转的转角和轴向时,其余四个边的方向直接表示传动轴中的旋转,这将产生所需的旋转。给出了一般情况下轴设置角的解析表达式。如果第一个和第三个轴是平行的,中间一个垂直于这些基准零(如四个圆圈状的衍射仪),那么这些减少θ1 =反正切(μ、σ)+(反正切(λ,v) -ψ½8π),θ2 = 2 s arcsin½(λ2 + v2),θ3 =(μ、σ)-(反正切(λ,v) -ψ½8π),s =±1 0≤arcsin(λ2 + v2)½≤½π,λ,μ,v和σ的四个组件旋转矢量构造λ,μ和v是旋转轴的方向余弦乘以sin1 / 2 θ对于一个旋转角θ, σ是cos 1 / 2 θ。ψ是一个常数,由λ和v测量方向的选择决定。一般情况下的结果也用更常规的变量来表示。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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