Characterizing the Coordinate Functions of Space Filling Curves

Abstract

The coordinate functions, \(f\) and \(g\), of a space filling curve are continuous functions from \([0,1]\) to \([0,1]\) so that \(F(t)=(f(t),g(t))\) maps \([0,1]\) onto the unit square. In the paper Coordinate Functions of Space Filling Curves written by Foran, several necessary conditions for a continuous function \(f\) are given for there to be a continuous \(g\) so that \(F(t)=(f(t),g(t))\) maps \([0,1]\) onto \([0,1]^2\). In this paper, we define a new condition for \(f\) that is both necessary and sufficient to assure that \(f\) has a matching coordinate function \(g\) such that \(F(t)=(f(t),g(t))\) fills the square.