Geometric shape parameterization of multidimensional finite non-regular point sets

The paper is devoted to a formal approach to parameterization the geometric shape of spatial point irregular sets in multidimensional Euclidean space. The geometric shape of the point set is described by means of the set of numerical parameters that determine the relative position of all points of the set up to similarity. Numeric shape parameters are defined as ratios of distances between given points and vertices of some basis simplex. The basis simplex is selected at the given set of points. It is shown that geometric shape of the set or the set of numerical parameters does not depend on the choice of the basis simplex. Change of the basis means recalculation of shape parameters values by linear transformation formulas. Amultidimensional object, the geometric shape of which is supposed to be described by numerical parameters, is previously subjected to simplicial approximation. Such an approximation is considered as a simplicial complex having a combinatorial structure. The paper also shows that all values of the shape parameters have to be satisfy some conditions that generate some area in the multidimensional space of parameters. To illustrate the described approach several examples are given. The proposed approach can be applied to solving a number of theoretical and applied problems of engineering geometry in the field of multidimensional geometric morphometry.