## ABSTRACT #

After decades of using only one map projection, the Polyconic, for its mapping program, the U.S. Geological Survey (USGS) now uses several of the more common projections for its published maps. For larger scale maps, including topographic quadrangles and the State Base Map Series, conformal projections such as the Transverse Mercator and the Lambert Conformal Conic are used. Equal-area and equidistant projections appear in the National Atlas. Other projections, such as the Miller Cylindrical and the Van der Grinten, are chosen occasionally for convenience, sometimes making use of existing base maps prepared by others. Some projections treat the Earth only as a sphere, others as either ellipsoid or sphere.

The USCS has also conceived and designed several new projections, including the Space Oblique Mercator, the first map projection designed to permit mapping of the Earth continuously from a satellite with low distortion. The mapping of extraterrestrial bodies has resulted in the use of standard projections in completely new settings. Several other projections which have not been used by the USGS are frequently of interest to the cartographic public.

With increased computerization, it is important to realize that rectangular coordinates for all these projections may be mathematically calculated with formulas which would have seemed too complicated in the past, but which now may be programmed routinely, especially if aided by numerical examples. A discussion of appearance, usage, and history is given together with both forward and inverse equations for each projection involved.

## INTRODUCTION #

The subject of map projections, either generally or specifically, has been discussed in thousands of papers and books dating at least from the time of the Greek astronomer Claudius Ptolemy (about A.D. 150), and projections are known to have been in use some three centuries earlier. Most of the widely used projections date from the 16th to 19th centuries, but scores of variations have been developed during the 20th century. In recent years, there have been several new publications of widely varying depth and quality devoted exclusively to the subject. In 1979, the USGS published Maps for America, a book-length description of its maps (Thompson, 1979). The USGS has also published bulletins describing from one to three projections (Birdseye, 1929; Newton, 1985).

In spite of all this literature, there was no definitive single publication on map projections used by the USGS, the agency responsible for administering the National Mapping Program, until the first edition of Bulletin 1532 (Snyder, 1982a). The USGS had relied on map projection treatises published by the former Coast and Geodetic Survey (now the National Ocean Service). These publications did not include sufficient detail for all the major projections now used by the USGS and others. A widely used and outstanding treatise of the Coast and Geodetic Survey (Deetz and Adams, 1934), last revised in 1945, only touches upon the Transverse Mercator, now a commonly used projection for preparing maps. Other projections such as the Bipolar Oblique Conic Conformal, the Miller Cylindrical, arid the Van der Grinten, were just being developed, or, if older, were seldom used in 1945. Deetz and Adams predated the extensive use of the computer and pocket calculator, and, instead, offered extensive tables for plotting projections with specific parameters.

Another classic treatise from the Coast and Geodetic Survey was written by Thomas (1952) and is exclusively devoted to the five major conformal projections. It emphasizes derivations with a summary of formulas and of the history of these projections, and is directed toward the skilled technical user. Omitted are tables, graticules, or numerical examples.

In USGS Bulletin 1532 the author undertook to describe each projection which has been used by the USGS sufficiently to permit the skilled, mathematically oriented cartographer to use the projection in detail. The descriptions were also arranged so as to enable a lay person interested in the subject to learn as much as desired about the principles of these projections without being overwhelmed by mathematical detail. Deetz and Adams’ (1934) work set an excellent example in this combined approach.

While Bulletin 1532 was deliberately limited to map projections used by the USGS, the interest in the bulletin has led to expansion in the form of this professional paper, which includes several other map projections frequently seen in atlases and geography texts. Many tables of rectangular or polar coordinates have been included for conceptual purposes. For values between points, formulas should be used, rather than interpolation. Other tables list definitive parameters for use in formulas. A glossary as such is omitted, since such definitions tend to be oversimplified by nature. The reader is referred to the index instead to find a more complete description of a given term.

The USGS, soon after its official inception in 1879, apparently chose the Polyconic projection for its mapping program. This projection is simple to construct and had been promoted by the Survey of the Coast, as it was then called, since Ferdinand Rudolph Hassler’s leadership of the early 1800’s. The first published USGS topographic “quadrangles,” or maps bounded by two meridians and two parallels, did not carry a projection name, but identification as “Polyconic projection” was added to later editions. Tables of coordinates published by the USGS appeared in 1904, and the Polyconic was the only projection mentioned by Beaman (1928, p. 167).

Mappers in the Coast and Geodetic Survey, influenced in turn by military and civilian mappers of Europe, established the State Plane Coordinate System in the 1930’s. This system involved the Lambert Conformal Conic projection for States of larger east-west extension and the Transverse Mercator for States which were longer from north to south. In the late 1950’s, the USGS began changing quadrangles from the Polyconic to the projection used in the State Plane Coordinate System for the principal State on the map. The USGS also adopted the Lambert for its series of State base maps.

As the variety of maps issued by the USGS increased, a broad range of projections became important: The Polar Stereographic for the map of Antarctica, the Lambert Azimuthal Equal-Area for maps of the Pacific Ocean, and the Albers Equal-Area Conic for the National Atlas (USGS, 1970) maps of the United States. Several other projections have been used for other maps in the National Atlas, for tectonic maps, and for grids in the panhandle of Alaska. The mapping of extraterrestrial bodies, such as the Moon, Mars, and Mercury, involves old projections in a completely new setting. Perhaps the first projection to be originated within the USGS is the Space Oblique Mercator for continuous mapping using imagery from artificial satellites.

It is hoped that this expanded study will assist readers to understand better not only the basis for maps issued by the USGS, but also the principles and formulas for computerization, preparation of new maps, and transference of data between maps prepared on different projections.