Minimizing Root-Mean-Square Linear Distortion in Common Conformal Map Projections

Abstract

The upcoming State Plane Coordinate System of 2022 has spurred considerable interest in design, development, and implementation of large-scale conformal map projections designed at Earth’s surface instead of the reference ellipsoid surface. Such projections address overall linear distortion, which accounts for the combination of scale distortion and ellipsoid height distortion. A method for minimizing root-mean-square (RMS) linear distortion in map projection design is presented. It is based upon least-squares “best fits” of map projection surfaces to Earth’s surface, represented by sets of height data points. Linear distortions, across three common conformal map projections (Lambert conformal conic, transverse Mercator, and Hotine oblique Mercator), are controlled by sets of “critical” parameters that have nonlinear functional relationships with position. These functional relationships are linearized and used in iterative least-squares solutions to find estimates for the parameters that minimize the sum of the squares of linear distortions for the entire input height data point set. The methodology can include weighting for priorities such as population and transportation corridors. Examples are presented and comparisons with existing map projections are made, for both unweighted and population-weighted data sets. It is recognized that there can be design considerations, involving linear distortion, other than or in addition to minimizing its root-mean square.