A geometric centroid principle and its application

A new concept of approximation for rigid point sets is suggested. As a nece ssary condition of optimality, the principle of the conjoint centroid is pr oved: to achieve a best approximation, certain co-sets must conjoin their c entroids. The practical use of the centroid principle, and how it opens up a non-classical method of modelling various aspects of orientational disord er in crystals, is demonstrated. The principle is applied to the interpreta tion of density data, to the prediction of high-pressure conformations thro ugh qualitative simulations, and to the prediction and computation of disor dered sets of possible reorientation pathways which explain the shape of th e 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.