Discrete Quantization on Spherical Geometries: Explicit Models, Computations, and Didactic Exposition

This article presents a comprehensive and analytically explicit study of optimal discrete quantization on spherical geometries equipped with the geodesic metric. Focusing on highly symmetric configurations on the unit sphere S2, we investigate three explicit models of discrete uniform distributions and derive closed-form expressions for their optimal quantizers and corresponding mean square quantization errors. (I) For N equally spaced points on the equator, we obtain exact error formulas for both divisible and non-divisible cases n∤N, demonstrating that optimal Voronoi cells form contiguous arcs with midpoint representatives. (II) For two antipodally symmetric small circles at latitudes ±ϕ0, each with M longitudes, we prove a no-cross-circle Voronoi phenomenon, establish symmetry-preserving optimality, and derive finite-sum error formulas together with sharp curvature-dependent bounds and asymptotics. (III) For a single small circle at latitude ϕ0, we obtain analogous exact error formulas and show that curvature reduces distortion by a factor of cos2ϕ0, while preserving the n−2 decay rate. Across all models, we rigorously establish the “block midpoint principle”: optimal Voronoi cells on a circle are contiguous azimuthal blocks, and their optimal representatives are the corresponding azimuthal midpoints. Numerical tables and illustrative figures highlight curvature effects and compare divisible and non-divisible cases. An algorithmic appendix provides pseudocode and a small, commented Python implementation to facilitate reproducibility. Written with didactic clarity while maintaining full mathematical rigor, this work bridges geometric intuition and analytic precision, providing explicit benchmark models that illuminate curvature effects and support further developments in quantization on curved manifolds.