Conformal mapping at elevation with implications for low-distortion projections

Abstract

This article develops the theory of the conformal mapping of the surface of constant height above the ellipsoid (ChS), which has implications for the design of low-distortion projections. This paper gives a development of the Mercator, polar stereographic, Lambert conformal conic (LCC) and transverse Mercator (TM) projections based on the ChS, a better Earth model than the reference ellipsoid (RE) for positions at height. This article provides new formulas for the LCC and TM inverse scale problems – when given a value for the local scale function, where does it occur? These formulas apply to both traditional mapping on the RE and novel mapping on the ChS.

Keywords:

• Low-distortion projections
• Conformal mapping
• Map-projection theory
• Mapping at height

Acknowledgements

The authors acknowledge the helpful and constructive comments from the anonymous reviewers.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Additional information

Notes on contributors

Craig M. Rollins

Craig M. Rollins retired from the geodesy department of the National Geospatial-Intelligence Agency in 2022 where he developed algorithms for map projections and supported cartographers in their need for accurate coordinates and suitable map projections. He has a M.A. degree in Mathematics from the University of Maryland (1976).

Thomas H. Meyer

Dr. Thomas H. Meyer is a Professor of Geodesy in the Department of Natural Resources and the Environment at the University of Connecticut. He published an undergraduate textbook on geodesy, and his peer-reviewed articles have focused on geometric geodesy, geodetic surveying and statistics.