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

TL;DR

This paper introduces a universal framework and encoding system for constructing and analyzing 2x2 space-filling curves, allowing detailed study of their structures, transformations, and recursive properties with efficient point location methods.

Contribution

It presents a novel, flexible construction framework for 2x2 space-filling curves without requiring self-similarity, along with a comprehensive encoding system and theoretical analysis tools.

Findings

Unified encoding for all curve forms and transformations

Deterministic methods for curve generation and transformation

Efficient point location within linear time complexity

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.