Equivalent Metric Coordinate Systems in the Background Space of Class 𝔻�

The work is devoted to a formal description of coordinate systems and their equivalence. The notion of a metric coordinate system is introduced, which includes a function , a metric tensor , and a metric (the length of the shortest path). An equivalence relation between systems is defined via a converter that preserves the metric, and a virtual space is defined as an equivalence class. The background space (the class containing the Cartesian system) is introduced. Lattices (intrinsic and converted) in coordinate systems are constructed. Eleven examplesof coordinate systems are considered: Cartesian, cylindrical, spherical, elliptic, parabolic cylindrical, parabolic, bipolar, toroidal, quasi-conical, conical, and ellipsoidal. For each system, the metric, the conversion (from Cartesian and back), and the lattices are presented