This book presents the most condensed information about the theory of distortion theory developed by N.A. In finding an ideal projection using the Airy criterion for an arbitrary mapping region is solved by the variational method using the Euler–Ostrogradsky system of equations under natural boundary conditions.

Introduction.- Map projections and their distortionsMap projections and their distortions.- The problem of finding the best projections.- Ideal projection according to the Airy criterion.- The best projection from a set of close-to-equal-area projections.- Airy projection.- Gauss–Kruger projection.- Arithmetic mean principle for the Gauss–Kruger projection.- Ideal and best projections according to Chebyshev’s criterion.- Appendixes.

This book presents the most condensed information about the theory of distortion theory developed by N.A. Tissot. It considers some of the issues of this theory to finding the best projections. Various criteria for ideal projections are analyzed. In finding an ideal projection using the Airy criterion for an arbitrary mapping region is solved by the variational method using the Euler–Ostrogradsky system of equations under natural boundary conditions. The same method is applied to a set of projections in which the sum of the extremal scale factors is equal to 2. It is shown that for these projections, the area distortions are quantities of the second order of smallness, while the linear distortions are quantities of the first order of smallness. The problem of finding the best projections using the Chebyshev criterion has been studied. Airy, Postel, Gauss–Kruger, and Markov projections are considered in detail.