بهزاد ملکان

The mathematical aspect of cartographic mapping is a process which establishes a unique connection between points of the earth’s sphere and their images on a plane. It was proven in differential geometry that an isometric mapping of a sphere onto a plane with all corresponding distances on both surfaces remaining identical can never be achieved since the two surfaces do not possess the same Gaussian curvature. In other words, it is impossible to derive transformation formulae which will not alter distances in the mapping process. Cartographic transformations will always cause a certain deformation of the original surface. These deformations are reflected in changes of distances, angles and areas. One of the main tasks of mathematical cartography is to determine a projection of a mapped region in such a way that the resulting deformation of angles, areas and distances are minimized. It is possible to derive transformation equations which have no deformations in either angles or areas. These projections are called conformal and equiareal, respectively. Since the transformation process will generally change the original distances it is appropriate to adopt the deformation of distances as the basic parameter for the evaluation of map projections. In 1861 an English astronomer, G. B. Airy, made the first significant attempt in cartography to introduce a qualitative measure for a combination of distortions. His measure of quality was designed to be an equivalent to the variance in statistics. A more realistic evaluation of the deformations at a point was suggested by German geodesisit, W. Jordan, in 1896. In 1959, Kavraisky recommended a small modification of the mean square deformations of Airy and Jordan by the logarithmic definition of linear deformation. Such altered mean square deformation are called Airy-Kavraisky and Jordan-Kavraisky. Using the above two mentioned criterions we can compute the mean square deformation of distances at a point. The evaluation and comparison of map projections of a closed domain is done by integration of the above two criterions. In this paper the first measure was used as the qualitative measure of map projections. The two criterions should lead to similar results but the application of the Airy-Kavraisky criterion in the computation process is much simpler. This is the main reason for its selection as the basis of finding the best projection. Optimization process was done in irregular domain of Iran for Lambert conic, Mercator cylindrical and stereographic azimuthal conformal projections. At first a grid composed of 165 points was created in the region. The scale factor was computed for the center of grid elements. The boundaries consist of a series of discrete points. Since the optimization domain is not regular like a spherical trapezoid, spherical cap or a hemisphere, so the minimization of the criterion leads to a least squares adjustment problem. For the Lambert conformal conic projection the optimization process will determine four unknown parameters: the geographic coordinates of metapole (φ0, λ0) and the projection constants C1 and C2. For other map projections the number of unknown parameters is three (φ0, λ0, C). The following table shows the numerical results of Airy-Kavraisky criterion after optimization. In this table shows the Airy-Kavraisky criterion before optimization and shows this criteria after optimization. Computational results show a decrease in Airy-Kavraisky criterion after optimization. This study of optimization of cartographic projections for small scale mappings was conducted to investigate the general approaches for obtaining the best projections using the Airy-Kavraisky measure of quality.

The mathematical aspect of cartographic mapping is a process which establishes a unique connection between points of the earth’s sphere and their images on a plane. It was proven in differential geometry that an isometric mapping of a sphere onto a plane with all corresponding distances on both surfaces remaining identical can never be achieved since the two surfaces do not possess the same Gaussian curvature. One of the main tasks of mathematical cartography is to determine a projection of a mapped region in such a way that the resulting deformation of angles, areas and distances are minimized. Most large-scale national topographic maps are based on conformal map projections such as transverse Mercator and Lambert conformal conic projections. The essential condition in every conformal map-projection is the infinitesimal similarity. Chebyshev studied conformal map projections, using the oscillation of the logarithm of the infinitesimal scale function as a measure of distortion. Chebyshev’s criterion states that the conformal map projection on Ω with minimum distortion is characterized by the property that the infinitesimal-scale σ is constant along the boundary of Ω. The oscillation in Ω of the logarithm of the infinitesimal-scale function associated to this best Chebyshev conformal map projection (or simply Chebyshev projection) will be called the minimum conformal distortion associated with Ω. Then we consider how to quantify the minimum conformal distortion associated with geographical regions. The minimum possible conformal mapping distortion associated with Ω coincides with the absolute value of the minimum of the solution of a Dirichlet boundary-value problem for an elliptic partial differential equation in divergence form and with homogeneous boundary condition. If the first map is conformal, the partial differential equation becomes a Poisson equation for the Laplace operator. The Dirichlet BVP could be solved by the finite element method (FEM). The FEM method is a procedure used in finding approximate numerical solutions to BVPs/PDEs. It can handle irregular boundaries in the same way as regular boundaries. It consists of the following steps to solve the elliptic PDE: 1- Discretize the (two-dimensional) domain into subregions such as triangular elements, neither necessarily of the same size nor necessarily covering the entire domain completely and exactly. 2- Specify the positions of nodes and number them starting from the boundary nodes and then the interior nodes. 3- Define the basis/shape/interpolation functions for each subregion. As a particular case, we consider the region of Iran in this paper. Conformal mapping equation in this region is solved for Mercator as the base map projection. To solve this equation three approaches are used: Finite Element Method (using Matlab Partial Differential Equation, PDE, Toolbox for square domain and Femlab code for arbitrary irregular domain), Fourier Method and Harmonic Polynomials. At the end, graphs associated with logarithm of the infinitesimal-scale function and also obtained results for coefficients of harmonic polynomials associated with the best Chebyshev projection over the region of Iran are presented.The minimum conformal distortion associated with square boundary domain estimated as 9.232×10-3 using finite element method and 9.243×10-3 using Fourier method. Also for the region of Iran with real domain, the value of this quantity estimated as 2.381×10-3 using finite element method and 2.462×10-3 using harmonic polynomials. Computations show that the results of three approaches are very close to each other. So for determination the best Chebyshev’s projection for a geographic region, the three mentioned approaches give the same results.