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

Abstract

We present an analytically explicit study of optimal discrete quantization on spherical geometries equipped with the geodesic metric, focusing on highly symmetric configurations on the unit sphere ${\mathbb{S}}^2$. Three discrete uniform models are analyzed and closed-form expressions for optimal quantizers and mean-square errors are derived. (I) For $N$ equally spaced points on the equator, exact error formulas are obtained for both divisible and non-divisible cases, showing that optimal Voronoi cells form contiguous arcs with midpoint representatives. (II) For two antipodally symmetric small circles at latitudes $\pm \ddot{I}{•}_0$, each with $M$ longitudes, we establish a no-cross-circle Voronoi phenomenon, symmetry-preserving optimality, and finite-sum error formulas with curvature-dependent bounds and asymptotics. (III) For a single small circle at latitude $\ddot{I}{•}_0$, analogous formulas are proved and curvature is shown to reduce distortion by a factor ${\cos }^2\ddot{I}{•}_0$ while preserving the $n^{-2}$ decay rate. Across all models we rigorously formulate the block-midpoint principle: optimal Voronoi cells are contiguous azimuthal blocks whose representatives are azimuthal midpoints. These explicit benchmark models clarify curvature effects and support further developments in quantization on curved manifolds.