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

Mrinal Kanti Roychowdhury

TL;DR

This work addresses optimal quantization for probability measures supported on one-dimensional spherical curves by adopting intrinsic geometry on the sphere. It develops a pedagogical framework based on geodesic distance $d_G$, Fréchet means, and spherical Voronoi regions, and shows that intrinsic quantization on a geodesic curve of length $L$ reduces to a classical one-dimensional problem, yielding $V_n = \frac{L^2}{12n^2}$ with equally spaced codepoints. The main contributions include explicit continuous and discrete models for great circles, small circles, and geodesic arcs, together with exact results for the equator and latitude parallels, plus a unifying principle and worked examples. These insights provide a clear geometric pathway to quantization on manifolds, with practical implications for directional statistics, spherical coding, and geometric data analysis. The treatment emphasizes geometric intuition and step-by-step derivations, making intrinsic sphere quantization accessible to beginning researchers while connecting to broader quantization theory.

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.

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

TL;DR

This work addresses optimal quantization for probability measures supported on one-dimensional spherical curves by adopting intrinsic geometry on the sphere. It develops a pedagogical framework based on geodesic distance , Fréchet means, and spherical Voronoi regions, and shows that intrinsic quantization on a geodesic curve of length reduces to a classical one-dimensional problem, yielding with equally spaced codepoints. The main contributions include explicit continuous and discrete models for great circles, small circles, and geodesic arcs, together with exact results for the equator and latitude parallels, plus a unifying principle and worked examples. These insights provide a clear geometric pathway to quantization on manifolds, with practical implications for directional statistics, spherical coding, and geometric data analysis. The treatment emphasizes geometric intuition and step-by-step derivations, making intrinsic sphere quantization accessible to beginning researchers while connecting to broader quantization theory.

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 -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 , the optimal -means form a uniform partition and the quantization error satisfies . 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.

Paper Structure

This paper contains 40 sections, 10 theorems, 105 equations.

Table of Contents

  1. Introduction and Geometric Preliminaries
  2. Great Circle, Small Circle, and Spherical Triangle
  3. Equator and the Prime Meridian, Latitude and Longitude.
  4. Relationship between three different coordinates: Cartesian, Spherical, and Geographical
  5. Geodesic Distance via Geographical Coordinates
  6. Arc Length on a Spherical Surface
  7. The two-argument arctangent function atan2
  8. Spherical Linear Interpolation (Slerp)
  9. Fréchet Mean
  10. Quantization Error on the Sphere
  11. Spherical Voronoi Regions
  12. Optimal Quantizers on the Sphere
  13. Extrinsic centroid condition and its relation to intrinsic (Fréchet) means
  14. Quantization Dimension and Quantization Coefficient
  15. Applications
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $C \subset \mathbb{S}^2_\rho$ be a smooth closed geodesic curve of total intrinsic length $L$ (e.g. a great circle or a small circle), and let $P$ be the uniform probability measure on $C$ with respect to arc-length. Then the intrinsic quantization problem on $(C,d_G)$ with squared geodesic dist

Theorems & Definitions (41)

  • Theorem 1.1: Conceptual reduction to the one-dimensional uniform case
  • Proposition 1.8.1
  • proof
  • Remark 1.8.2
  • Example 1.9.1: Euclidean space
  • Example 1.9.2: Spherical space
  • Proposition 3.1: Centroid condition; heuristic form
  • Remark 3.2
  • Remark 3.3
  • Theorem 4.2.1: Equator: structure of optimal $n$-means
  • ...and 31 more