Modern equations of diffractometry. Goniometry

The various settings of a goniostat can best be calculated using a coordinate-free abstract operator notation. Concepts such as free axes, signed axes, angles and finite rotations are defined here using modern geometrical methods, and virtualized methods of describing combinations of rotations and of solving goniometric equations are given. These have the advantages of simplifying analysis and of being applicable to all types of machine. Three practical examples appropriate to the use of an area-detector diffractometer are presented: the synthesis of true precession motions using the concept of a 'virtual goniostat'; the generation of convenient crystal-viewing positions; and the solution of the equation of diffraction for the generalized, non-normal beam, rotation method. Algorithmic solutions are quoted in all three cases, corresponding to the code used on the FAST system at Cambridge and to that released to the EEC Cooperative Workshop on Position-Sensitive Detector Software in Paris in 1986. The emphasis has been placed on simple and reliable methods of computation. In each case, the equations reduce to the same, fundamental, form, containing four behaviourally distinct terms: one variable, one invariant, one odd and one even. The classical quatemion notation of theoretical physics, motor algebra and applications of dual numbers are also discussed.