Calculating Map Projections for the Ellipsoid

Application of standard map projections to the ellipsoidal Earth is often considered excessively difficult. Using a few symbols for frequently-used combinations, exact equations may be shown in compact form for ellipsoidal versions of conformal, equal-area, and equidistant projections developed onto the cone, cylinder (in conventional position), and plane, as well as for the polyconic projection. Series are needed only for true distances along meridians. The formulas are quite interrelated. The ellipsoidal transverse and oblique Mercator projections remain more involved. An adaptation of the Space Oblique Mercator projection provides a new ellipsoidal oblique Mercator which, unlike Hotine's, retains true scale throughout the length of the central line.