Two different two-dimensional quadrilateral meshes are compared. Both of them are asymptotically optimal quadrilateral meshes for bilinear approximation. In the paper interesting error properties of two types of meshes are presented. Both types of mesh are generated by a coordinate transformation of a regular mesh of squares. The coordinate transformation is derived by integrating the Hessian matrix of a data function as a metric tensor. The first type of meshes has an equidistributed error and the maximum interpolation error is asymptotically same over all elements. The second type of mesh has fast super-convergence error property for certain saddle-shaped data. The second type of mesh may be an order of magnitude more accurate than the first type.