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

Authors

• Mrinal Kanti Roychowdhury, The University of Texas Rio Grande ValleyFollow

Document Type

Article

Publication Date

1-1-2026

Abstract

This expository paper provides a unified and pedagogical introduction to optimal quantization for probability measures supported on spherical curves and discrete subsets ofthe sphere, emphasizing both continuous and discrete settings. We first present a detailedgeometric and analytical foundation for intrinsic quantization on the unit sphere, includingdefinitions of great and small circles, spherical triangles, geodesic distance, Slerp interpolation,the Fréchet mean, spherical Voronoi regions, centroid conditions, and quantizationdimensions. Building upon this framework, we develop explicit continuous and discretequantization models on spherical curves, namely great circles, small circles, and greatcircular arcs—supported by rigorous derivations and pedagogical exposition. For uniformcontinuous distributions, we compute optimal sets of n-means and the associated quantizationerrors on these curves; for discrete distributions, we analyze antipodal, equatorial,tetrahedral, and finite uniform configurations, illustrating convergence to the continuousmodel. The central conclusion is that for a uniform probability distribution supported on aone-dimensional geodesic subset of total length L, the optimal n-means form a uniformpartition and the quantization error satisfies (Formula presented.).The exposition emphasizesgeometric intuition, detailed derivations, and clear step-by-step reasoning, making it accessibleto beginning graduate students and researchers entering the study of quantization onmanifolds. This article is intended as an expository and tutorial contribution, with the mainemphasis on geometric reformulation and pedagogical clarity of intrinsic quantization onspherical curves, rather than on the development of new asymptotic quantization theory.

Comments

© 2025 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license.

Recommended Citation

Roychowdhury, Mrinal Kanti. "Optimal Quantization on Spherical Surfaces: Continuous and Discrete Models—A Beginner-Friendly Expository Study." Mathematics 14, no. 1 (2025): 63. https://doi.org/10.3390/math14010063

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Publication Title

Mathematics

DOI

10.3390/math14010063