Classimetry of Polygonal Chains (CMPC). English edition. Chapter 1. The Alphabet of Directions

Version: Ch-1(ver-1). Individual chapter under construction. This is the first 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. Chapter 1 introduces the anisotropic alphabet of directions A, consisting of eight symbols: four axial directions (→, ↑, ←, ↓) and four sectoral directions (↗, ↖, ↙, ↘). Each symbol corresponds to an equivalence class of non-zero vectors in the plane based on the signs of their coordinates. Key concepts introduced in this chapter:- path code (P-code) as a tuple of edge directions;- discrete stability of the path code under small deformations;- cell in the parameter space;- elementary R- and R̃-functions (with an explanation of the notation change from the Russian edition);- hierarchy and self-similarity of the R-functional construction;- minimality of the eight-direction alphabet;- equivalence classes of one-segment polylines;- connection with Bézier curves and B-splines. 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).