Minimization of Distortion of Cartographical Maps

Abstract

One of the most perplexing problems confronting Cartographers has been the minimization of distortion associated with the mapping of surfaces from the sphere to the plane. In this thesis the objective is to obtain a conformal map of an open, simply connected region of the world (regarded as a sphere) which minimizes the linear or areal distortion (magnification) in transformed mappings at points in the region. | To achieve this result, an apparently little known theorem of Tchebychef [9, pp. 242] has been employed, which states that the foregoing will be the case when the magnific ation on the boundary of the region to be mapped is constant THEOREM 1. If U is a simply connected region bounded by a twice differentiable curve, then there exists one and up to a similarity transformation of E, only one conformal map projection which minimizes this ratio sup o/inf o. This "best possible" conformal map projection is characterized by the property that its infinitesimal-scale function a(x) is constant along the boundary of U.