# Quantization for uniform distributions on hexagonal, semicircular, and elliptical curves

## TL;DR

This paper studies optimal quantization for uniform distributions on various geometric curves, providing exact formulas, algorithms, and analyzing quantization dimensions and coefficients.

## Contribution

It introduces methods to determine optimal quantization sets and errors for uniform distributions on hexagons, semicircular discs, and ellipses, including exact formulas and algorithms.

## Key findings

- Exact formulas for n-means when n=6k for hexagons.
- Quantization dimension equals the geometric dimension.
- Finite positive quantization coefficient established.

## Abstract

In this paper, first we have defined a uniform distribution on the boundary of a regular hexagon, and then investigated the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$. We give an exact formula to determine them, if $n$ is of the form $n=6k$ for some positive integer $k$. We further calculate the quantization dimension, the quantization coefficient, and show that the quantization dimension is equal to the dimension of the object, and the quantization coefficient exists as a finite positive number. Then, we define a mixture of two uniform distributions on the boundary of a semicircular disc, and obtain a sequence and an algorithm, with the help of which we determine the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$ with respect to the mixed distribution. Finally, for a uniform distribution defined on an elliptical curve, we investigate the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$.