John P. Snyder

A Comparison of Pseudocylindrical Map Projections

There are at least 80 published pseudocylindrical map projections, with straight parallels and curved meridians. Of these, about 40 are equal area (a few of which are not significantly different from others), and about 20 have equidistant parallels. Many are mentioned only qualitatively if at all in references other than the original sources. Some 40 inventors are involved. Seven significantly different projections using sinusoids offer alternatives, such as flat or pointed poles, equal area, or equidistant parallels. A dozen elliptical pseudocylindricals offer these variations, perhaps more attractively, with added possibilities from full versus partial ellipses. Five parabolics are rounder and probably preferable aesthetically to their sinusoidal counterparts. Nine projections with hyperbolas or converging straight lines for meridians are almost useless. Over two dozen projections use less well-known curves in attempts to give maps less real or apparent distortion. They include seven pseudocylindricals ...

Labeling Projections on Published Maps

To permit accurate scaling on a map, and to use the map as a source of accurate positions in the transfer of data, certain parameters—such as the standard parallels selected for a conic projection—must be stated on the map. This information is often missing on published maps. Three current major world atlases are evaluated with respect to map projection identification. The parameters essential for the projections used in these three atlases are discussed and listed. These parameters should be stated on any map based on the same projection.

“Magnifying-Glass” Azimuthal Map Projections

For maps focusing on a region of interest, but including surrounding areas to provide a setting, new azimuthal projections have been developed with a “magnifying-glass” effect. On two such projections, inside a circle bounding the region of interest is a standard Azimuthal Equidistant or Lambert Azimuthal Equal-Area projection. Between this circle and an outer bounding circle, azimuths remain true and the radial or area scale, respectively, remains constant, but at a reduced value. On four other projections, the inner portion is a standard azimuthal projection, which may be Stereographic, Gnomonic, or the above, but beyond this portion, the radial scale is gradually reduced to zero. Equivalents with rectangular boundaries are also available.

World Maps With Natural Boundaries

The graticule of meridians and parallels is a largely artificial type of map boundary that can detract from the display of irregular features such as oceans. Such natural boundaries as shorelines may be used instead as the boundary of world maps. The principle of natural boundaries has been applied to several examples of equal-area or conformal world ocean maps with single or multiple lobes. By careful selection of the poles and centers, these maps can show both oceans and continents in their entirety on a single map.

Minimum-Error Equal-Area Map Projections

An equal-area map projection minimizing scale variation within a desired region may be prepared by combining the principle of least squares with an iteration of three elementary equal-area transformations. Constants are determined on a one-time basis using the Gram-Schmidt orthonormalization procedure with any number of chosen points on the Earth sphere or ellipsoid for which minimal distortion is desired. A standard initial projection such as the Lambert Azimuthal Equal-Area is then transformed using these constants and the iterative transformations to produce the final equal-area projection with irregular lines of constant distortion suiting the chosen region.

New equal-area map projections for noncircular regions

A series of new equal-area map projections has been devised. Called Oblated Equal-Area, its lines of constant distortion follow approximately oval or rectangular paths instead of the circles of the Lambert Azimuthal Equal-Area projection or the straight lines of the Cylindrical Equal-Area projection. The projection series permits design of equal-area maps of oblong regions with less overall distortion of shape and scale than equal-area maps on other projections.

An Innovative World Map Projection

A map projection suitable for many thematic maps of the world not involving area-related data is an oblique orthographic projection of E. N. Gilberts Two-World Conformal Globe, centered at 5° N. latitude and 5° E. longitude. Although the result is neither conformal nor equal-area, it provides a rounded view of almost the entire world, essentially uninterrupted.

A Low-Error Conformal Map Projection for the 50 States

The author has developed a conformal map projection showing all 50 States of the United States and the passages connecting them with a scale distortion of less than ±2 percent, considerably less than the +12-percent, −3-percent range of the present USGS 50-State map. This includes all islands of Alaska and Hawaii, although the central portion of the north Pacific has a scale error of 3.2 percent. Because of the small variation, ellipsoidal corrections are also usefully incorporated. The derivation employs least-squares and standard conformal transformations using complex algebra to determine suitable coefficients for a complex polynomial. Forward and inverse formulas are given for the final projection.

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 Hotines, retains true scale throughout the length of the central line.

Map-Projection Graphics from a Personal Computer

Some inexpensive personal computers may be programmed to produce, at a nominal incremental cost, map projection graphics useful as educational tools. Programs have been developed to produce outline maps based on anyone of dozens of projections, in almost any aspect, at a size of up to 13.5 by 27.1 cm or 20.3 by 20.3 cm [5⅓by 10⅔ in. or 8 by 8 in.]. They are printed in a dot-matrix format normally containing up to 320 by 768 dots, with alternatives of 640 by 768 or 960 by 576 dots. Available options include features such as interrupted projection, miscellaneous great or small circles, and Tissot indicatrices. The maps often require many hours to prepare, but the programs can run unattended after initial parameters have been entered.