A Comparison of Variational Projection and Cartographic Projection by Ritz’s Method

Abstract

The application of alternative mathematical methods in creating cartographic projections is an interesting factor, which affects the optimization of distortions and their distribution in the projected territory. This article presents the methodology for the creation and comparison of conformal cartographic projections formed by alternative mathematical methods of minimizing the integral criterion for scale distortion in Slovakia. The creation of the variational projection is based on the Airy-Kavraiskii criterion of evaluating the projection on the displayed area by solving Laplace's equation. The second projection is created by solving Poisson's equation using Ritz's method. Our analysis showed that the variational projection of Slovakia achieves more satisfactory distortion values than the cartographic projection created using Ritz's method. The advantage of Ritz's method is that it is possible to choose a boundary condition for a predefined undistorted convex closed curve. In this paper, we have also derived specific members of the map equations for cartographic projection based on solving Poisson's equation by Ritz's method.

Keywords

• cartographic projection
• variational criterion
• laplace's equation
• poisson's equation
• the dirichlet integral
• ritz's method