Application of Nelder-Mead nonlinear optimization method for minimizing map projection distortion – demostrated on an azimuthal projection

Abstract

The search for map projections with least possible distortion, satisfying selected criteria which integrate different measures of distortion, is one of the more important tasks of cartography. In the nineteenth century, many integral based criterions have been proposed, minimization of which is considered as achieving an optimal distortion pattern for a given projection. In the present time of mass computerization and constantly rising computation speed, popularity of numerical solutions of the mentioned criteria has risen. These numerical solutions are achieved by application of nonlinear optimization methods.
A nonlinear function minimization method proposed by Nelder and Mead (Nelder and Mead, 1965) was used to optimize map projections of the spherical globe for small scale mapping by Frank Canters (2002). Canters optimized projections of the whole globe, for which flat coordinates were given by fifth order polynomials. Parameters of these polynomials were either longitude and latitude on the globe or flat coordinates of a given parent projection. The objective function was the revised Peters measure of distortion (Canters, 2002), which is a finite distortion measure comparing distance between two given points on the globe with their distance on the map, for a large set of randomly chosen points.
In the present study, Nelder-Mead algorithm is used to minimize distortion of an azimuthal projection of the sphere in the normal aspect, so that it will satisfy Airy’s criterion. The obtained solution will be then compared with an analytical-strict solution for this criterion, as given by Gdowski (1967). The parallel radius in the formulas describing flat coordinates of the optimized projection is written as a linear combination of the parent projections radius and a power series of , which denotes spherical distance from the north pole. Optimized variables will be the coefficients of the said linear combination with flat coordinates of the optimized projection. As the objective function, we consider the value of adequately modified Peters measure, so that its minimization corresponds with the minimization of the sum of the squares of the errors of scale factors in principal directions, for the given region, as stated in the Airy’s criterion.
The author plans to continue his research of nonlinear optimization methods for other projections and distortion minimization criterions.