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Space-Filling Curves

Shihan Kanungo

TL;DR

Space-filling curves provide continuous surjections from a one-dimensional interval to higher-dimensional spaces, challenging intuition about curves. The paper reviews, formalizes, and compares Hilbert's (a simple Peano variant) and Lebesgue's space-filling curves, and presents the Hahn-Mazurkiewicz theorem as a classification of attainable sets as continuous images of $[0,1]$. It constructs the Hilbert curve via a dyadic correspondence that preserves adjacency and proves key properties, including a Hölder condition with exponent $1/2$ and measure-preservation on subintervals. It also discusses the general theorem that any compact, connected, weakly locally-connected set is a continuous image of $[0,1]$, and emphasizes practical applications such as Google's $S2$ cells and GPU locality-preserving codes. The work highlights the deep connections between topology, fractal constructions, and real-world spatial indexing.

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.

Space-Filling Curves

TL;DR

Space-filling curves provide continuous surjections from a one-dimensional interval to higher-dimensional spaces, challenging intuition about curves. The paper reviews, formalizes, and compares Hilbert's (a simple Peano variant) and Lebesgue's space-filling curves, and presents the Hahn-Mazurkiewicz theorem as a classification of attainable sets as continuous images of . It constructs the Hilbert curve via a dyadic correspondence that preserves adjacency and proves key properties, including a Hölder condition with exponent and measure-preservation on subintervals. It also discusses the general theorem that any compact, connected, weakly locally-connected set is a continuous image of , and emphasizes practical applications such as Google's cells and GPU locality-preserving codes. The work highlights the deep connections between topology, fractal constructions, and real-world spatial indexing.

Abstract

We examine space-filling curves, which are surjective continuous maps from to some higher-dimensional space, usually the unit square . 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 that are the image of a space-filling curve. Finally, we discuss real-world applications of Hilbert curves, in particular Google's Cells.
Paper Structure (8 sections, 9 theorems, 5 equations, 5 figures)

This paper contains 8 sections, 9 theorems, 5 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Basic concepts
  3. Peano's curve
  4. Lebesgue's curve
  5. Main results
  6. Properties of the Hilbert curve
  7. The Hahn-Mazurkiewicz Theorem
  8. Applications

Key Result

Proposition 2.2

Chains of quartic intervals satisfy the following properties:

Figures (5)

  • Figure 1: The Dragon Fractal
  • Figure 2: The Hilbert Curve
  • Figure 3: The Lebesgue Curve
  • Figure 4: Construction of $\phi$
  • Figure 5: Google's $S2$ cells

Theorems & Definitions (13)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Theorem 3.1
  • Lemma 3.2
  • Definition 4.1
  • ...and 3 more