Local modification of the geometric shape of planar point sets

   The article presents a local modification method and an algorithm of the irregular point sets configuration in the Euclidean plane. Irregular point set is a set of vertices of some planar figure, specifically it becomes a broken line. The figure can be a base-set for some smooth planar peace-nonlinear curve. Numerical parameters or coordinates of the base-setare variable by given intervals, as well as the shape of the base-set. The local modification does not modify such metrical parameters of the base-set as length of sections and total length of the broken line. The authors introduce the concept of the admissible parametric local modification of the broken line. The admissible parametric local modification depends upon the number of variable points. In order to demonstrate the local modification the authors propose the hierarchic structuring of the base-set that means successive selection such sub-sets of adjacent points, which allow realizing the local modification of various number of parameters. The 0-, 1- and 2-parametric modifications are described in detail. The article shows two examples in order to illustrate the method. The first example describes the local modification of piece-linear segments of non-linear energy rays reflector. The aim of the segments modification is to change the direction of reflected rays. Local modification is implemented in light-guiding structure, in which flat section of reflecting surface consists of broken with fixed lengths of links. Discrete modifications of reflecting surface generate stepped distribution of energy illumination at receiver. The other example presents one of possible realizations and modification of the base-set for some accidental shape of some surface transverse section. The transverse section is a composite curve that simulates the section of a freely hanging fabric.

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