Classimetry of Polygonal Chains (CMPC). English edition. Chapter 3. Hybrid R/Rtilde Descriptions of Polygonal Contours

Chapter 3 is under construction. This chapter addresses a fundamental limitation of the classical theory of R-functions, developed by V.L. Rvachev, when applied to the classification of polygonal contours. The canonical R₀ system, while perfectly suited for describing two-dimensional sectors, fails for axial directions (rays) due to the topological impossibility of representing a one-dimensional ray as the positivity set of a continuous function — a problem referred to as the ray problem. To overcome this, the chapter introduces regularized Rtilde-functions with a parameter delta > 0, which provide smooth approximations of axial directions while preserving the limit transition to the ideal ray as delta -> 0. A hybrid construction is then proposed, combining classical R-functions for sectoral directions with regularized Rtilde-functions for axial directions within a single conjunction. The resulting hybrid R/Rtilde-function serves as a characteristic function of equivalence classes of polygonal contours: it is positive inside a class, zero on its boundaries, and negative outside it. The chapter also discusses the choice of the regularization parameter delta, its adaptive selection, and the compatibility properties of the hybrid construction at sector boundaries. The Russian edition of the monograph was published in 2026 (DOI: 10.5281/zenodo.19781575).The English edition is being prepared chapter by chapter under the same open access license (CC BY 4.0).