Space-Filling Curves

TL;DR

This paper explores the mathematical properties of space-filling curves like Peano and Lebesgue, discusses their theoretical foundations such as the Hahn-Mazurkiewicz theorem, and highlights practical applications like Google's S2 Cells.

Contribution

It provides a comprehensive overview of space-filling curves, their key properties, and their relevance in real-world applications, connecting theory with practice.

Findings

Characterization of space-filling curves via the Hahn-Mazurkiewicz theorem

Properties of Peano's and Lebesgue's curves

Application of Hilbert curves in Google's S2 Cells

Abstract

We examine space-filling curves, which are surjective continuous maps from $[0,1]$ to some higher-dimensional space, usually the unit square $[0,1]^2$. In particular, we define Peano's curve and Lebesgue's curve, and state some of their properties. We also discuss the Hahn-Mazurkiewicz theorem, which characterizes those subsets of $\mathbb{R}^n$ that are the image of a space-filling curve. Finally, we discuss real-world applications of Hilbert curves, in particular Google's $S2$ Cells.