Four Methods for Low-Distortion Projections

Low-distortion projections (LDPs) produce mapping grids formulated so that distances obtained by inversing grid coordinates match as best as possible the equivalent horizontal distances observed at elevation. The need for LDPs arises in places at high elevations, such as in the mountainous western region of the United States, where the enlargement of distances due to elevation can exceed mapping accuracy tolerances. This paper discusses four methods for constructing LDPs-they are either new interpretations of established methods or new outright. The four methods are analyzed by or built upon two assumptions that are new in the context of LDPs: (1) the notion of horizontal distances at elevation needs a rigorous definition; and (2) a surface of constant height above the reference ellipsoid (RE) provides such a definition. Regarding the second assumption, the more general notion is an elevated reference surface, and several other candidates are admitted, but this paper chooses the constant-h surface (ChS) to be the elevated reference surface and the means to build and analyze LDPs. The four methods are successive improvements on the agenda to construct a conformal projection of the ChS and are analyzed in turn. LDP Method 2 is promoted as the solution most likely to meet users' requirements for simplicity and geodetic integrity. Not too surprisingly, these are competing requirements.