Optimization of Lambert, Mercator and Stereographic map projections for the Iranian territory using Airy-Kavraisky criterion

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 (?0, ?0) and the projection constants C1 and C2. For other map projections the number of unknown parameters is three (?0, ?0, 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.