Quasi-isometric parametrization of a curvilinear quadrangle and a metric of constant curvature (Q1920129)

A mapping \(u=u(s,t)\), \(v=v(s,t)\) is called quasi-isometric if the ratio of the distance between two points \((s_1,t_1)\) and \((s_2,t_2)\) to the distance between their images \((u(s_1,t_1))\), \(v(s_1,t_1))\) and \((u(s_2,t_2)\), \(v(s_2,t_2))\) is bounded from above and below, i.e.: \[ 0<\sigma_1 \leq {\sqrt{(u(s_2,t_2)-u(s_1,t_1))^2+(v(s_2,t_2)-v(s_1,t_1))^2}\over \sqrt{(s_2-s_1)^2+(t_2-t_1)^2}} \leq \sigma_2. \] Quasi-isometry implies quasi-conformality. The authors study in this paper the construction of a grid in a curvilinear quadrangle \(G\) onto a surface of constant curvature by means of a quasi-isometric mapping from the unit square and a five-parametric family of metrics on the unit square.