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Lebesgue curve is a space-filling curve that fills the unit square throughlinear interpolation. In this study, we generalise Lebesgue's construction togenerate space-filling curves from any given planar substitution satisfying amild condition. The generated space-filling curves for some known substitutionsare elucidated. Some of those substitutions further induce relatively densefractal-like sets in the plane, whenever some additional assumptions are met.

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