Map projections used for large-scale quadrangles by the U.S. Geological Survey

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tf? U.S. Geological Survey Circular 982

Supersedes USGS Circular 57, State Coordinates and Polyconic Maps, dated May 1949, from which some portions are adapted By John P. Snyder

Supersedes USGS Circular 57, State Coordinates and Polyconic Maps, dated May 1949, from which some portions are adapted

1. Map-projection. I. Title. II. Series: Geological Survey circular ; 982. GA110.S58 1986 526.8 86-600139

Free on application to the Books and Open-File Reports Section, U.S. Geological Survey, Federal Center, Box 25425, Denver, CO 80225

The Earth as an ellipsoid ................................................. . Conformal map projections ................................................. . The Polyconic projection .................................................. . The Transverse Mercator projection ........................................ . The Lambert Conformal Conic projection .................................... . Comparison between quadrangles of different projections ................... . Distortion of the map material; grid overlays ............................. . References

Figure 1. Diagrams showing development of respective projections and type and location of principal distortions .................... 3 2. Diagrams showing matching of quadrangle sheets and zones on the respective projections ................................. 4

2. Diagrams showing matching of quadrangle sheets and zones on the respective projections ................................. CONTENTS PEge

iii MAP PROJECTIONS USED FOR LARGE-SCALE QUADRANGLES BY THE U.S. GEOLOGICAL SURVEY

After the U.S. Geological Survey (USGS) was created in 1879, detailed large-scale mapping of the country soon became one of its primary objectives. Until the late 1950's, only the Polyconic projection was used for the primary USGS mapping product, i.e., large-scale quadrangle maps. new quadrangles was changed to the Lambert Conformal Conic or the Transverse Mercator, which had been adopted by the Coast and Geodetic Survey in the 1930's for the State Plane Coordinate System (SPCS). The development of standardized zones based upon the Universal Transverse Mercator (UTM) grid and projection system le~ to USGS use of this form of the Transverse Mercator projection for some more recent quadrangles. Although one SPCS zone in Alaska is based on the Oblique Mercator projection, this projection is not used for the USGS quadrangles.

For all large-scale quadrangle mapping by the USGS, the Earth is treated as an ellipsoid or spheroid, a sphere flattened by about 1 part in 300. All descriptions below refer to the ellipsoidal forms of the three map projections used. The USGS is currently undertaking a change in the datum upon which the mapping occurs. This involves a change in the dimensions of the ellipsoid as well as a change of the position of its polar and equatorial axes with respect to the continental land masses. A shift from one datum to another results in a slight change of latitude and longitude for all points on a map, as well as a different change in the positions of grid coordinates. For this reason, the notation on USGS quadrangles stating "North American Datum 1927" or "1983" is one of the essential parameters of the map projection that defines these maps.

The two most important properties that may be preserved with map projections are area and shape, but no one projection can preserve both. Area-preserving projections are called equal-area or equivalent and are used for many geographical aprlications. Most modern topographic mapping throughout the world is plotted using conformal map projections. On a conformal map, each small element is basically correct in shape. Angles at each point are correct, and consequently the local scale in every direction around any one point is constant, so the map user can measure distance and direction between near points with a minimum of difficulty. Conformal maps may also be prepared in practice by fitting together small piecer of other conformal maps that have been enlarged or reduced, whereas pieces of

Publication authorized by the Director, u.s. Geological Survey, on January 27, 1986.

By John P. Snyder

In the 1950's, the projection for m~st

THE EARTH AS AN ELLIPSOID

CONFORMAL MAP PROJECTIONS nonconformal projections would require reshaping as well. Because there is no angular distortion, all meridians intersect parallels at right angles on a conformal projection, just as they do on the Earth. Standard lines may be specified for a conformal map to eliminate scale and area distortion along these lines and to minimize distortion elsewhere. On conformal maps the size of USGS large-scale quadrangles, the distortion of scale and area is very small.

About 1820, the easily constructed Polyconic projection began to be pronoted as the basis of large-scale mapping. The USGS used this projection for the earliest quadrangles, changing only in the 1950's to other projections, although relabeling the map legend has lagged considerably behind the change. The Polyconic is neither conformal nor does it preserve correct area. For 7.5- and 15-minute quadrangles, however, the distortion is insignificant. Along the straight cen~ral meridian, the projection is free of distortion. Each parallel is also true to scale, but the other meridians are too long, and constantly change scale.

The parallels of latitude are circular arcs spaced at their true distanr.es along the central meridian, but with radii equal to the length of the element of a cone tangent at the particular parallel. The projection receives its name from the fact that each cone is different (fig. 1). Meridians are marked on each parallel at their true distances from the central meridian, but the meridians are theorotically complex curves connecting these points. Lines of constant scale in a north-south direction run roughly parallel to the central meridian, but they are curved.

The location of the central meridian on a Polyconic quadrangle is norma~.ly at the center of the map. Frequently, however, a bounding meridian has been used as the central meridian. For 15- and 7.5-minute quadrangles, the difference rosulting from positioning the central meridian in the center or on the edge is neglitible. Furthermore, because of the limited coverage, the meridians were drawn straight rather than curved. In any case, adjacent quadrangles match perfectly from north to south, and because of the straight meridians they also match from east to w~st, but they cannot be assembled beyond a single row or single column without gaps (fig. 2). The projection is not recommended for maps of considerable east-west extent nor for any new maps in view of other projections available.

THE TRANSVERSE MERCATOR PROJECTION

The most frequently used conformal projection for large-scale mapping i~ the Transverse Mercator. The Transverse Mercator is considered a cylindrical pr~jection because the Earth may be conceptually (but not geometrically) projected by ~rapping a cylinder around a globe so that the cylinder is tangent along a meridian, or secant along lines parallel to this meridian (fig. 1). When the meridians ~nd parallels are properly placed on this cylinder, and the cylinder is cut alor~ a line perpendicular to this meridian, unrolled, and laid flat, the Transverse Mercator projection results. The central meridian is a straight line. All other meridians and parallels on U.S. quadrangles are theoretically complex curves, but are practically straight segments. The central meridian has a constant scale, but this is usually reduced from the nominal map scale to balance errors in measurement over the rest of the map. The lines of constant scale are nearly straight lines parallel to the central meridian. When the scale factor along the central meridian is r~duced, there are two lines of true scale that are symmetrical with respect to the central meridian. The projection was developed by Lambert and later Gauss during the late 18th and early 19th centuries, but it was almost ignored until the 20th cent.ury, when it was adopted for much of the topographic mapping in Europe under the name

1 11 1 II 1.-- cylinder

TRANSVERSE LAMBERT CONFORMAL CONIC PROJECTION POLYCONIC PROJECTION MERCATOR PROJECTION

Figure 1.--Diagrams showing development of respective projections and type and location of principal distortions.

N N

TRANSVERSE MERCATOR PROJECTION

Figure 2.--Diagrams showing matching of quadrangle sheets and zones on the respective projections.

Figure 2.--Diagrams showing matching of quadrangle sheets and zones on the respective projections. DISTORTION OF THE MAP MATERIAL; GRID OVERLAYS

Of much greater practical significance than the difference between projections on large-scale quadrangles is the fact that distortions may be developed ir a map through the physical instability of the medium on which it is drawn or prirted. It is well known that paper will shrink or expand with changes in the surrouneing atmospheric conditions, as well as from other causes. The change in dimenfions of any sheet of paper will normally not be the same in all directions, and chsnges of shape as well as of size may result. These changes can be so great as to c~literate completely the theoretical distortions and scale changes resulting from th€ map projection. It is not unusual for a dimensional change as great as 1/8 inch to occur in a single quadrangle map sheet on map paper. Under such conditionf, it is impossible to assemble perfectly a number of adjoining sheets, each with it.s own unpredictable paper distortion, regardless of the projection on which the ~~ps were prepared.

Special plastic films are available in which distortions can be reduce~ or controlled, and such improved materials should be used, when feasible, in t.he preparation of maps. Unless materials of exceptional stability are employed, it. may be assumed that the theoretical differences between the projections for large-scale mapping will always be less than the probable distortions of the paper or other materials on which the map may be produced. Since the late 1950's, large-rcale USGS maps have been prepared on stable-base materials, and should be obtained ir this form for precision mapping.

Since paper maps will continue to be commonly used, the use of gridlin~s on maps is especially helpful in overcoming the effects of expansion and contraction. A square grid based on UTM coordinates for every 1,000 meters is typically slown on 7.5-minute USGS quadrangles. Since this grid expands with the paper, coor~inates of intermediate points may be determined with a high degree of precision, and then converted to geographic coordinates or to distances from other points.

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