Minimum conformal mapping distortion according to Chebyshev’s principle: a case study over peninsular Spain

Abstract

Defining the distortion of a conformal map projection as the oscillation of the logarithm of its infinitesimal-scale σ, Chebyshev’s principle states that the best (minimum distortion) conformal map projection over a given region Ω of the ellipsoid is characterized by the property that σ is constant on the boundary of that region. Starting from a first map of Ω, we show how to compute the distortion δ0(Ω) of this Chebyshev’s projection. We prove that this minimum possible conformal mapping distortion associated with Ω coincides with the absolute value of the minimum of the solution of a Dirichlet boundary-value problem for an elliptic partial differential equation in divergence form and with homogeneous boundary condition. If the first map is conformal, the partial differential equation becomes a Poisson equation for the Laplace operator. As an example, we compute the minimum conformal distortion associated with peninsular Spain. Using longitude and isometric latitude as coordinates, we solve the corresponding boundary-value problem with the finite element method, obtaining δ0(Ω)=0.74869×10−3. We also quantify the distortions δl and δutm of the best conformal conic and UTM (zone 30) projections over peninsular Spain respectively. We get δl=2.30202×10−3 and δutm=3.33784×10−3.