An Introduction to Space-Filling Curves and

Their Applications

Kyle Byrne 1407493

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Contents

1 Introduction 31.1 Notation and Definitions . . . . . . . . . . . . . . . . . . . . . . . 4

2 Peano’s Space-Filling Curve 52.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Geometric Representation and Approximating Polygons . . . . . 5

3 Hilbert’s Space-Filling Curve 53.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Geometric Representation and Approximating Polygons . . . . . 5

4 Sierpinski’s Space-Filling Curve 54.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.3 Geometric Representation and Approximating Polygons . . . . . 5

5 Applications of Space-Filling Curve’s 55.1 Scientific Computing . . . . . . . . . . . . . . . . . . . . . . . . . 55.2 Geometric Data Structures . . . . . . . . . . . . . . . . . . . . . 5

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Abstract

This paper should be interpreted as an introduction to the conceptof space filling curves. Looking at examples created by Peano, Hilbertand Sierpinski. Such curves have motivation in computer science as anefficient way to traverse multi-dimensional data.

1 Introduction

We start with a definition fundamental to geometry.

Definition 1. An n-dimensional manifold is a topological space M for whichevery point x ∈M has a neighbourhood homeomorphic to Euclidean space En.

That is for each point inM we can find a neighbourhood X such that thereexist’s a continuous mapping

f : En → X

Such that the inverse f−1 is also continuous.If we describe these invertible maps from En to each of our subspaces X as a

chart, with the collection of them being an atlas. We can consider the transitionmaps betwen any two charts, that is the composition of one chart with theinverse of another. If the transition maps for every chart are differentiable thenthe manifold M is said to be a differentiable manifold. further specialisationof this is the smooth manifold for which every transition map is C∞, that isthe derivatives of all orders exist. Examples of such smooth manifolds are theclosed unit-hypercubes (That is the n-dimensional analogue of the unit square),the Mobius Strip, or even circles and lines in 2 dimensions.

In 1878 the German mathematician George Cantor proved that for any twofinite-dimensional smooth manifolds there existed a bijective mapping from oneto the other. That is the cardinalities of any two finite-dimensional smoothmanifolds is equal. This has a rather remarkable result that the interval [0, 1]can be mapped bijectively onto the square [0, 1]× [0, 1]. In fact it is quite simpleto construct such a bijection.

Theorem 1. There exists a bijection f : [0, 1]→ [0, 1]2

Proof. We represent (x, y) ∈ [0, 1]2\{(0, 0)} by it’s unique infinite decimal ex-pansion.

(x, y) = (0.x1x2x3 . . . , 0.y1y2y3 . . .)

Such that we dont end in a string of infinite zeros, i.e ∀N ∈ N,∃p, q > N such that xp 6=0 and yp 6= 0. In this way 0.36 would be represented 0.359999 . . ..

Then consider the function

f(x, y) =

{0 if (x,y)=(0,0)

0, x1y1x2y2x3y3 . . . otherwise

Then clearly as each decimal expansion is unique this is an injective functionwhich by the Schroeder-Bernstein theorem implies that |[0, 1]2| ≤ |[0, 1]|, but as[0, 1] ⊂ [0, 1]2 then we also have that |[0, 1]| ≤ |[0, 1]2|. Hence |[0, 1]| = |[0, 1]2|and so there must exist a bijection from [0, 1→ [0, 1]2]

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The immediate question after this result is wether such a mapping can becontinuous , it was Netto in 1879 who showed this couldnt be true.

Theorem 2 (Netto). If f represents a bijective map from an m-dimensionalsmooth manifold µn onto an n-dimensional smooth manifold µn and m 6= nthen f is necessarily discontinuous.

The next progression was to consider whether it would be possible to finda continuous surjective mapping, losing the condition of bijectivity. In 1890Peano constructed the first such curve, these curves are now refered to as spacefilling curves.

1.1 Notation and Definitions

For the purpose of this text we will use the following letters to denote commonsubsets of Rn

I = [0, 1] the closed unit interval

Q = [0, 1]2 the closed unit square

T the closed triangular region with vertices (0,0),(2,0) and (1,1)

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2 Peano’s Space-Filling Curve

2.1 Definition

2.2 Properties

2.3 Geometric Representation and Approximating Poly-gons

3 Hilbert’s Space-Filling Curve

3.1 Definition

3.2 Properties

3.3 Geometric Representation and Approximating Poly-gons

4 Sierpinski’s Space-Filling Curve

4.1 Definition

4.2 Properties

4.3 Geometric Representation and Approximating Poly-gons

5 Applications of Space-Filling Curve’s

5.1 Scientific Computing

5.2 Geometric Data Structures

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