Map Projection Equations

Abstract

Abstract : Map projection equations in cartesian coordinates are derived for the most important mapping schemes. The important unifying principles of differential geometry are applied to produce the equal area, conformal and conventional projections. This report has collected under one cover the major map projections useful to scientists and engineers. The notation is uniform. The derivations proceed in all cases from first principles to usable equations suitable for hand plotting, digital/analog plotting, or CRT display. The problem is stated, and the reader is introduced to the terminology of the art of map projections. Basic transformation theory is introduced, and then particularized for the transformation from the spheroid or sphere onto a developable surface. The criterion for the derivations is to use the most simple and direct approach. The model of the earth is then considered. The most recent parameters to describe the figure of the earth are given, and tables incorporating these are included for meridian length, parallel length and the relation between geodetic and geocentric latitude. The computer programs which generated these tables are included in the appendix. Equal area, conformal, and conventional projection equations are derived. These equations are incorporated into an original computer program which generated the map plotting tables for the most important projections. This program, which produces either a complete grid or individualized points, is also in the appendix. Since the proof of all of the derivations is a correct graticule of meridians and parallels, original figures of these have been produced. The plotting tables and the figures reflect the modern parameters for the earth.