Problem of the optimal cartographic projections

Cartographic projections are basis for the graphical representation of various territories in small scale mapping. Proper selection of projection reduces the deformation of the presented territory, which is bounded by a boundary line. In most cases, this border line is not a mathematically defined curve, which is most easily displayed in the form of a closed polygon. The optimal cartographic projections based on a selected criterion of quality are those whose constants lead to the smallest value of the criterion. In the presented work it is recommended to use Airy-Kavrajski criterion whose minimization is actually minimization of the second Euclidean norm. The solution of optimal projections of various classes is reduced to the method of least squares. Fast modern computers enable the optimization of an arbitrary territory by evaluating the selected criterion in a finite number of points.