Optimal Quantization on Spherical Surfaces: Continuous and Discrete Models -- A Beginner-Friendly Expository Study

TL;DR

This paper offers a clear, pedagogical introduction to optimal quantization on spherical surfaces, covering both continuous and discrete models, with explicit calculations and geometric insights suitable for beginners.

Contribution

It provides the first comprehensive, beginner-friendly exposition of intrinsic quantization on spheres, including explicit models, derivations, and convergence analysis for various distributions.

Findings

Optimal n-means form uniform partitions on spherical curves.

Quantization error for uniform distributions scales as L^2/(12n^2).

Discrete configurations converge to continuous models with increasing n.

Abstract

This expository paper provides a unified and pedagogical introduction to optimal quantization for probability measures supported on spherical curves and discrete subsets of the sphere, emphasizing both continuous and discrete settings. We first present a detailed geometric and analytical foundation for intrinsic quantization on the unit sphere, including definitions of great and small circles, spherical triangles, geodesic distance, Slerp interpolation, the Frechet mean, spherical Voronoi regions, centroid conditions, and quantization dimensions. Building upon this framework, we develop explicit continuous and discrete quantization models on spherical curves, namely great circles, small circles, and great circular arcs supported by rigorous derivations and pedagogical exposition. For uniform continuous distributions, we compute optimal sets of $n$-means and the associated quantization errors on these curves; for discrete distributions, we analyze antipodal, equatorial, tetrahedral, and finite uniform configurations, illustrating convergence to the continuous model. The central conclusion is that for a uniform probability distribution supported on a one-dimensional geodesic subset of total length $L$, the optimal $n$-means form a uniform partition and the quantization error satisfies $V_n = L^2/(12n^2)$. The exposition emphasizes geometric intuition, detailed derivations, and clear step-by-step reasoning, making it accessible to beginning graduate students and researchers entering the study of quantization on manifolds. This article is intended as an expository and tutorial contribution, with the main emphasis on geometric reformulation and pedagogical clarity of intrinsic quantization on spherical curves, rather than on the development of new asymptotic quantization theory.