Two geometric characteristic of quasicircles
Two variations of the Ahlfors three points characterization for quasicircles are presented. The first states that a Jordan curve \(C\) is a quasicircle (the image of a circle under a quasiconformal mapping of the extended plane) iff for some constant \(K\) for any four points \(z_1\), \(z_2\), \(w_1\), \(w_2\) on \(C\) there exists a \(K\)-quasiconformal mapping \(f\) mapping \(C\) onto itself, \(f(\infty)=\infty\) and \(f(z_i)= w_i\), \(i= 1,2\). Another characterization states that being a quasicircle is equivalent to a property called bounded circular distortion. The latter condition means that there is \(0< c< 1\) such that for all \(x\) in \(C\) and \(r> 0\) the disk \(B(x,cr)\) meets only the \(x\)-component of \(C\cap B(x,r)\).
scientific article; zbMATH DE number 2138465