Riemmanian metric of harmonic parametrization of geodesic quadrangles and quasi-isometric grids

We consider the problem of generating a 2D structured boundary-fitting rectangular grid in a curvilinear quadrangle D with angles αi=ϕi+π/2αi=ϕi+π/2, where −π/2<ϕi<π/2−π/2<ϕi<π/2, i=1,…,4i=1,…,4. We construct a quasi-isometric mapping of the unit square onto D; it is proven to be the unique solution to a special boundary-value problem for the Beltrami equations. We use the concept of “canonical domains”, i.e., the geodesic quadrangles with the angles α1,…,α4α1,…,α4 on surfaces of constant curvature K=4sin⁡(ϕ1+ϕ2+ϕ3+ϕ4)/2K=4sin⁡(ϕ1+ϕ2+ϕ3+ϕ4)/2, to introduce a special class of coefficients in the Beltrami equations with some attractive invariant properties. In this work we obtain the simplest formula representation of coefficients gjkgjk, via a conformally equivalent Riemannian metric of harmonic parametrization of geodesic quadrangles. We also propose a new, more robust method to compute the metric for all parameter values.