A method for the generation of quasi-isometric boundary-fitted curvilinear coordinates for arbitrary domains is developed on the basis of the quasi-isometric mappings theory and conformal representation of spherical and hyperbolic geometries. A one-parameter family of Riemannian metrics with some attractive invariant properties is analytically described. We construct the quasi-isometric mapping between the regular computation domain $cal R$ and a given physical domain $cal D$ that is conformal with respect to the unique metric from the proposed one-parameter class. The identification process of the unknown parameter takes into account the high parametric sensitivity of metrics to the parameter. For this purpose we use a new technique for finding the geodesic quadrangle with given angles and a conformal module on the surface of constant curvature, which makes the method more robust. The method allows more direct control of the grid cells size and angle over the field as the grid is refined. Illustrations of this technique are presented for the case of one-element airfoil and several test domains.