Planar Substitutions to Lebesgue type Space-Filling Curves and Relatively Dense Fractal-like Sets in the Plane

Ismail Ozkaraca

doi:10.1016/j.jmaa.2023.127654

Planar Substitutions to Lebesgue type Space-Filling Curves and Relatively Dense Fractal-like Sets in the Plane

Ismail Ozkaraca

Royal (Dick) School of Veterinary Studies

Research output: Contribution to journal › Article › peer-review

Abstract

We present an algorithm for generating curves filling the unit square; i.e. space-filling curves, from any given planar substitution satisfying a mild condition. The proposed algorithm is mimicking construction steps of Lebesgue’s curve and is based on linear interpolation. Generated space-filling curves for some known substitutions are elucidated. Some of those substitutions further induce relatively dense fractal-like sets in the plane, whenever some additional assumptions are satisfied.

Original language	English
Article number	127654
Pages (from-to)	1-25
Number of pages	25
Journal	Journal of mathematical analysis and applications
Volume	530
Issue number	2
Early online date	18 Aug 2023
DOIs	https://doi.org/10.1016/j.jmaa.2023.127654
Publication status	Published - 15 Feb 2024

Keywords / Materials (for Non-textual outputs)

Space-filling curves
Lebesgue curve
Fractal-like set

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10.1016/j.jmaa.2023.127654

1-s2.0-S0022247X23006571-mainFinal published version, 1.21 MBLicence: CC BY

Cite this

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title = "Planar Substitutions to Lebesgue type Space-Filling Curves and Relatively Dense Fractal-like Sets in the Plane",

abstract = "We present an algorithm for generating curves filling the unit square; i.e. space-filling curves, from any given planar substitution satisfying a mild condition. The proposed algorithm is mimicking construction steps of Lebesgue{\textquoteright}s curve and is based on linear interpolation. Generated space-filling curves for some known substitutions are elucidated. Some of those substitutions further induce relatively dense fractal-like sets in the plane, whenever some additional assumptions are satisfied.",

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Planar Substitutions to Lebesgue type Space-Filling Curves and Relatively Dense Fractal-like Sets in the Plane

Abstract

Keywords / Materials (for Non-textual outputs)

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