The Axiata mapping problem and symmetry implications on cuspidal conchoids (Q509125)

The points \(x\) and \(-x\) on the real axis have the same distance from \(0\). The Axiata mapping problem is the problem of finding a mapping A from the \(x\)-axis into the real plane such that \((1,0)\) and \((0,1)\) are fixed and such that if \(P_1\) and \(P_2\) are the images of \((x,0)\) and \((1/x,0)\), respectively with \(x \in \mathbb{R} \backslash \{0\}\) and \(|x| \leqslant1\), then the corresponding distances from \(P_1\) and \(P_2\) to the point \((1,0)\) must be the same and its value must be either \(|1-x|\) or \(|1-(1/x)|\). The author gives two mappings solving the Axiata problem and he discusses their properties. One of them contains a cuspidal member of the family of conchoids of Nicomedes, the other contains a cuspidal member of the conchoids of a circle, namely, the cardioid.