Optimization of a polyconic projection for Poland with respect to Airy's criterion

Abstract

One of the fundamental tasks of mathematical cartography is to find projections with minimized total distortion. Airy's criterion demands that the sum of the squares of the principal scale errors should be minimized for the mapped area, so that the scale distortion in all directions is minimal. In this article, a polyconic projection for Poland, optimized with respect to Airy's criterion will be presented.
Three parametric models of the polyconic projection will be discussed. For a given number of parameters, these models will be optimized, using a modified Nelder and Mead nonlinear optimization algorithm. The modification consisted of extending Nelder and Mead algorithm with a mutation operator, known from evolutionary algorithms, which adds normally distributed random values to the projection parameters. This helped to prevent the algorithm from converging to a false minimum. Regularity of the optimized projections has been inspected and the distortion pattern has been illustrated using numerically interpolated lines of constant distortion.
For a limited number of the objective function calculations, chosen parametric models have been compared, with respect to the averagely achieved mean scale distortion , for many algorithm evaluations, for the mapped area The mean distortion of lengths is understood as the squared, averaged and square rooted length distortions in directions of extreme scales - it corresponds to the minimised measure of distortion according to Airy's criterion.