Classimetry of Polygonal Chains (CMPC). English edition. Chapter 2. Classical R-functions by V.L. Rvachev

Chapter 2 is under construction. This is the second chapter of the English edition of "Classimetry of Polygonal Chains" (CMPC). The work is being published chapter by chapter; the complete book will be assembled when all chapters are ready. The chapter provides a systematic exposition of the classical theory of R-functions (Rvachev functions) as applied to the classimetry of polygonal chains. The material is organised into three main sections. Section 2.1 introduces the R_0 system — a set of real functions closed under three fundamental operations: R-conjunction (∧⁰), R-disjunction (∨⁰), and R-negation (¬⁰). For each operation we give a precise definition and discuss its logical interpretation as an analogue of Boolean AND, OR, and NOT. We prove the key algebraic properties of these operations: commutativity, associativity, distributivity, De Morgan's laws, and idempotence. The section concludes with a demonstration of how any logical formula describing a geometric domain as a combination of half-spaces can be directly transformed into an R-function. Section 2.2 focuses on sectoral directions (↗, ↖, ↙, ↘). Using the R_0 apparatus, we define elementary R-functions for each direction as F_(↗)(x,y)=x∧⁰y, F_(↖)(x,y)=(-x)∧⁰y, F_(↙)(x,y)=(-x)∧⁰(-y), F_(↘)(x,y)=x∧⁰(-y). We prove that each such function is positive exactly inside the corresponding open quadrant and zero on the coordinate axes. An equivalent analytic representation via the identity min(x,y) = (x+y-|x-y|)/2 is provided, showing that these functions are piecewise linear and can be computed without conditional statements. Section 2.3 formulates the sign preservation principle — the cornerstone of the theory of R-functions. The theorem states that for any logical formula Φ built from elementary statements f_i > 0, after replacing logical connectives with R-operations, the sign of the resulting R-function coincides with the truth value of Φ at all points where the sum of the arguments is non-negative. This property guarantees that an R-function serves as a continuous sign-definite characteristic of a geometric domain: positive inside, zero on the boundary, negative outside. The significance of this principle for classimetry is discussed: stability under small parameter perturbations, differentiability inside cells, and the interpretation of R-values as a class metric.The chapter concludes with a summary and explicit examples demonstrating the construction of R-functions for two-segment polylines and the verification of sign preservation. 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).