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Construction, Transformation and Structures of 2x2 Space-Filling Curves

Zuguang Gu

TL;DR

The paper presents a universal expansion-based framework for constructing 2x2 space-filling curves, transcending strict self-similarity by expanding level-0 seeds into level-k curves via a complete set of level-0 to level-1 expansions. It introduces a robust symbolic-encoding system that assigns each curve a seed plus an expansion-code sequence, enabling deterministic generation, transformation, and reduction of curves, as well as precise arithmetic for locating points. The authors classify curve families by geometric and structural attributes, including corner-induced and side-induced shapes, and show how classic curves like the Hilbert curve and Moore/$\beta\Omega$-curves fit within this framework. They also analyze homogeneous vs. non-homogeneous forms, hierarchical shape generation, and the interplay between transformations, reductions, and encodings, culminating in a linear-time method to locate any point on the curve and a complete taxonomy of curve shapes. This framework provides a flexible, extensible toolkit for designing and analyzing 2x2 space-filling curves with broad implications for encoding, transformation, and geometric understanding of such curves.

Abstract

The 2x2 space-filling curve is a type of generalized space-filling curve characterized by a basic unit is in a "U-shape" that traverses a 2x2 grid. In this work, we propose a universal framework for constructing general 2x2 curves where self-similarity is not strictly required. The construction is based on a novel set of grammars that define the expansion of curves from level 0 (a single point) to level 1 (units in U-shapes), which ultimately determines all $36 \times 2^k$ possible forms of curves on any level $k$ initialized from single points. We further developed an encoding system in which each unique form of the curve is associated with a specific combination of an initial seed and a sequence of codes that sufficiently describes both the global and local structures of the curve. We demonstrated that this encoding system is a powerful tool for studying 2x2 curves and we established comprehensive theoretical foundations from the following three key perspectives: 1) We provided a deterministic encoding for any unit on any level and position on the curve, enabling the study of curve generation across arbitrary parts on the curve and ranges of iterations; 2) We gave deterministic encodings for various curve transformations, including rotations, reflections and reversals; 3) We provided deterministic forms of families of curves exhibiting specific structures, including homogeneous curves, curves with identical shapes, partially identical shapes, and with completely distinct shapes. We also explored families of recursive curves, subunit identically or differently shaped curves, completely non-recursive curves, symmetric curves and closed curves. Finally, we proposed a method to calculate the location of any point on the curve arithmetically, within a time complexity linear to the level of the curve.

Construction, Transformation and Structures of 2x2 Space-Filling Curves

TL;DR

The paper presents a universal expansion-based framework for constructing 2x2 space-filling curves, transcending strict self-similarity by expanding level-0 seeds into level-k curves via a complete set of level-0 to level-1 expansions. It introduces a robust symbolic-encoding system that assigns each curve a seed plus an expansion-code sequence, enabling deterministic generation, transformation, and reduction of curves, as well as precise arithmetic for locating points. The authors classify curve families by geometric and structural attributes, including corner-induced and side-induced shapes, and show how classic curves like the Hilbert curve and Moore/-curves fit within this framework. They also analyze homogeneous vs. non-homogeneous forms, hierarchical shape generation, and the interplay between transformations, reductions, and encodings, culminating in a linear-time method to locate any point on the curve and a complete taxonomy of curve shapes. This framework provides a flexible, extensible toolkit for designing and analyzing 2x2 space-filling curves with broad implications for encoding, transformation, and geometric understanding of such curves.

Abstract

The 2x2 space-filling curve is a type of generalized space-filling curve characterized by a basic unit is in a "U-shape" that traverses a 2x2 grid. In this work, we propose a universal framework for constructing general 2x2 curves where self-similarity is not strictly required. The construction is based on a novel set of grammars that define the expansion of curves from level 0 (a single point) to level 1 (units in U-shapes), which ultimately determines all possible forms of curves on any level initialized from single points. We further developed an encoding system in which each unique form of the curve is associated with a specific combination of an initial seed and a sequence of codes that sufficiently describes both the global and local structures of the curve. We demonstrated that this encoding system is a powerful tool for studying 2x2 curves and we established comprehensive theoretical foundations from the following three key perspectives: 1) We provided a deterministic encoding for any unit on any level and position on the curve, enabling the study of curve generation across arbitrary parts on the curve and ranges of iterations; 2) We gave deterministic encodings for various curve transformations, including rotations, reflections and reversals; 3) We provided deterministic forms of families of curves exhibiting specific structures, including homogeneous curves, curves with identical shapes, partially identical shapes, and with completely distinct shapes. We also explored families of recursive curves, subunit identically or differently shaped curves, completely non-recursive curves, symmetric curves and closed curves. Finally, we proposed a method to calculate the location of any point on the curve arithmetically, within a time complexity linear to the level of the curve.

Paper Structure

This paper contains 88 sections, 71 theorems, 273 equations, 30 figures, 20 tables.

Table of Contents

  1. Introduction
  2. Expansion rules
  3. Level 0-to-1 expansion
  4. Rotation
  5. Design of the expansion rules
  6. Expansion to level k
  7. Recursive expansion
  8. Corners
  9. Connect level-1 units
  10. Expansion code
  11. Example
  12. Expansion path
  13. Encode the curve
  14. The encoding system
  15. Special curves
  16. ...and 73 more sections

Key Result

Proposition 3.1

The form of $\mathcal{P}_k$ is only determined by the expansion of the first base in $\mathcal{P}_{k-1}$ ($k \ge 1$).

Figures (30)

  • Figure 1: Generation of 2x2 curves from level 0 to level 3. First row: generation under the copy-paste mode; Second row: generation under the expansion mode. The curves belong to a special form, the Hilbert curve.
  • Figure 2: The complete set of expansion rules from level 0 to level 1. Grey segments and arrows represent entry and exit directions of corresponding units.
  • Figure 3: Traversal on the level-1 unit of $I^{(1)}_1$.
  • Figure 4: Corners of a 2x2 unit. The lower left and upper right corners have values of 1 and the lower right and upper left corners have values of 2. The corner-tuple of the 2x2 unit is composed of the values of the entry and exit corners. The level-1 unit in this example is $R_1$.
  • Figure 5: Expansion paths of two curves. Top: the two expansion paths of $\mathcal{P}_1 = R^{90}|1$; Bottom: the two expansion paths of $\mathcal{P}_2 = R^{90}|11$. The meaning of $R^{90}|1$ and $R^{90}|11$ will be explained in later sections. The 2-tuple under each point is the corner-tuple for each base (rotation included).
  • ...and 25 more figures

Theorems & Definitions (173)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 4.1
  • Definition 4.2: Identical curves
  • Lemma 4.3
  • proof
  • Proposition 4.1
  • proof
  • ...and 163 more