Some problems of conformal mappings of spherical domains

Abstract

In this study the problem of finding the conformal mapping from a sphere onto a plane with a given scale function independent of longitude is solved for an arbitrary spherical domain. The obtained results are compared with the well-known projections used in cartography and geophysical fluid dynamics. The problem of minimization of the distortion under conformal mappings is solved for domains in the form of the spherical disk. The distortions of some extensively used conformal mappings are compared with the distortions of orthogonal mappings.

Comparison of different spatial grids for numerical schemes of geophysical fluid dynamics

“…2). For such a domain the most uniform conformal mapping is the stereographic projection tangent to the sphere at the centerpointP [3,5]. Its distortion coefficient is δ str Ω γ ,P = 2 1 + cos γ .…”

“…The result of existence and uniqueness of the solution was formulated and proved by Chebyshev and Grave: if Ω is a simply connected domain bounded by a twice differentiable curve, then there exists one, and, up to a similarity transformation of the plane, only one conformal mapping which minimizes the distortion coefficient δ (see [10,15]). The explicit form for the minimum distortion mapping was given in [3,5] for the spherical disk Ω γ ,P consisting of the points P = (a, λ, θ ) such that d S P , P aγ , where d S P , · is the spherical distance to the centerpointP = a,λ,θ and aγ (γ ∈ (0, π)) is the spherical radius (see Fig. 2).…”

“…can be formulated as the problem of solving the following system of PDEs for unknown functions x (λ, θ), y (λ, θ) [5,6]:…”

“…Our aim is to verify through numerical experiments up to what extent the theoretical results presented in [3][4][5] for domains of a specific geometry are applicable in the practical case of the rectangular computational domains. The paper is structured as follows.…”

On Conformal Conic Mappings of Spherical Domains

“…As a measure of the homogeneity of the computational grid one can use the ratio between the maximum and minimum values of the mapping factor over the considered domain: In particular, this criterion is suitable for generation of computational grids for explicit and semi-implicit schemes [ 1 , 7 , 8 ]. As far as we know, the use of the variation coefficient α for measuring the homogeneity of the computational grids was first proposed and studied in [ 1 ] and the further analysis of the properties of this coefficient and justification of its application in the atmosphere-ocean numerical models was performed in different works of the same authors (e.g., [ 4 , 7 , 8 ]).…”