This paper proposes an evaluation index Ln for map projection. Balance of area and angular distortion is considered in Ln. Though distortions around poles are often disregarded, Ln evaluates them by same method as distortions at other parts.
The index Ln is weighted mean of eSn and eAn over the globe, where eS and eA are values like strain of area and angle. Ln depends on map projection, n, and wS, which is weight for eSn. If n=2, minimization of Ln is an application of least square method.
This paper evaluates projections by range of n by which Ln converges; this method is named ‘order evaluation of distortion.’ It evaluates characteristics of map projection at poles, or generally at singular points, where theory of Tissot's indicatrix is not applicable. If distortion at poles shall not be disregard, map projection with the best order evaluation of distortion, it means ρ=κ=0, shall be used.
Values of L1，L2，and L3 are numerically calculated for some map projections. The result of the calculation and other consideration led a conclusion that L2 is appropriate for the evaluation of map projections. Aitoff transformation of stereographic projection has the smallest L2 values for about 70 % range of wS. The projection is therefore highly evaluated as balancing area and angular distortions over the world including poles.
