**Authors**

**Abstract**

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 ? with minimum distortion is characterized by the property that the infinitesimal-scale ? is constant along the boundary of ?. The oscillation in ? 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 ?.

Then we consider how to quantify the minimum conformal distortion associated with geographical regions. The minimum possible conformal mapping distortion associated with ? 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.

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 ? with minimum distortion is characterized by the property that the infinitesimal-scale ? is constant along the boundary of ?. The oscillation in ? 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 ?.

Then we consider how to quantify the minimum conformal distortion associated with geographical regions. The minimum possible conformal mapping distortion associated with ? 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.

**Keywords**