Airy criterion and the ideal mapping of an ellipsoid of revolution onto a sphere

Abstract

This paper addresses the problem of constructing an ideal projection of an ellipsoid of revolution onto a sphere based on the Airy criterion. General equations required to solve the problem are derived. In particular, the Euler–Urmaev system is obtained, allowing a clear illustration of Gauss's theorem that a distortion-free projection between these surfaces cannot exist. The Euler–Ostrogradsky system is also derived to find the projection that minimizes distortion according to the Airy criterion. Natural boundary conditions for the ideal projection are analyzed. It is shown that on the boundary of the mapping region, Tissot’s indicatrices are aligned either along the normals or tangents to the boundary, and one of the extremal linear scale factors is equal to unity. Since the value of the Airy criterion depends not only on the projection’s mapping functions but also on the radius of the sphere, an additional integral condition is introduced alongside the Euler–Urmaev system, the Euler–Ostrogradsky system, and the natural boundary conditions. According to this condition, the integral of the area distortion over the entire mapping region of the ideal projection must be equal to zero. Two specific cases are examined in detail: projection of the entire ellipsoid and of a region bounded by a parallel. For comparison, conformal projections optimized according to the Airy criterion were also constructed for the same mapping regions. The resulting ideal projections can be used in geodesy for solving direct and inverse geodetic problems, and in cartography for constructing double projections.