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

TL;DR

This paper provides explicit models and formulas for optimal discrete quantization on spherical geometries, analyzing symmetric configurations on the unit sphere with detailed error bounds and geometric properties.

Contribution

It introduces three explicit, symmetric quantization models on the sphere, deriving closed-form error formulas and revealing geometric principles like the block-midpoint rule.

Findings

Exact error formulas for equatorial points show optimal Voronoi arcs.

Finite-sum error bounds depend on curvature and asymptotics.

Curvature reduces distortion by a factor of cos^2(φ_0).

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 $\pm\phi_0$, 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 $\phi_0$, analogous formulas are proved and curvature is shown to reduce distortion by a factor $\cos^2\phi_0$ 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.