A geometric centroid principle and its application

A new concept of approximation for rigid point sets is suggested. As a necessary condition of optimality, the principle of the conjoint centroid is proved: to achieve a best approximation, certain co-sets must conjoin their centroids. The practical use of the centroid principle, and how it opens up a non-classical method of modelling various aspects of orientational disorder in crystals, is demonstrated. The principle is applied to the interpretation of density data, to the prediction of high-pressure conformations through qualitative simulations, and to the prediction and computation of disordered sets of possible reorientation pathways which explain the shape of the electron-density distribution reconstructed from diffraction experiments. It is also demonstrated how an inversion of the centroid principle can be used to model forces between the parts of the disordered structures.