  Lebesgue curve is a space-filling curve that fills the unit square through
linear interpolation. In this study, we generalise Lebesgue's construction to
generate space-filling curves from any given planar substitution satisfying a
mild condition. The 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 met.
