Simple plane arcs of positive area (Q1322574)

The authors construct for a given real number \(\beta\in [0,1[\) a simple arc \(f: [0,1]\Rightarrow \mathbb{R}^ 2\), which lies in the closed unit square but meets its boundary only in the corners \((0,0)\) and \((1,1)\), and satisfies for every Borel subset \(E\) of \([0,1[\), \(\lambda^ 2(F(E))= \beta\cdot \lambda(E)\), where \(\lambda^ 2\) and \(\lambda\) denote the Lebesgue measure of the plane resp. in the line. This property extends the constructions of such curves given by Knopp and Sierpiński.