A method is presented for automated best-matching alignment of threedimensional models represented by ensembles of points. A normalized spatial discrepancy (NSD) is introduced as a proximity measure between threedimensional objects. Starting from an inertia-axes alignment, the algorithm minimizes the NSD; the ®nal value of the NSD provides a quantitative estimate of similarity between the objects. The method is implemented in a computer program. Simulations have been performed to test its performance on model structures with speci®ed numbers of points ranging from a few to a few thousand. The method can be used for comparative analysis of structural models obtained by different methods, e.g. of high-resolution crystallographic atomic structures and low-resolution models from solution scattering or electron microscopy.

The problem of uniqueness of the low-resolution shape determination from small-angle scattering by isotropic monodisperse systems is considered. The particle shape is represented by the envelope function parameterized using spherical harmonics as described in a previous paper [Svergun & Stuhrmann (1991). Acta Cryst. A47, 736-744]. Computer simulations are made on the model bodies with sharp boundaries exactly represented by spherical harmonics. If the number of independent parameters describing the shape is 1 to 1.5 times the number of Shannon channels covered by the data set, the shape restoration is found to be unique and stable with respect to the random and systematic errors. The resolution limits of the straightforward shape determination are connected to the computational accuracy of the model intensities; with current algorithms, shapes described by 15 to 20 independent parameters can be uniquely determined. The results form a basis for an ab initio low-resolution shape determination in terms of spherical harmonics.

Practical aspects of low‐resolution shape determination in small‐angle scattering studies of biological macromolecules in solution are considered. The shape restoration method using spherical harmonics [Svergun, Volkov, Kozin & Stuhrmann (1996). Acta Cryst. A52, 419–426] is extended to account for deviations from the idealized model and to work directly on raw experimental data sets. An algorithm to restore the structure of homodimeric particles in terms of the shape of the monomer and the separation between the monomers is implemented. Applications of the program to the shape restoration of several proteins, with known and unknown crystal structures, from X‐ray solution‐scattering data are presented.