Transverse Mercator Projections and U.s. Geological Survey Digital Products

A common question about U.S. Geological Survey (USGS) digital raster graphics (DRG) is “how can I remove the map collars and join multiple quadrangles together?” The question does not have a simple answer because of the characteristics of USGS quadrangles. Quadrangle neatlines are defined by lines of latitude and longitude, so quadrangles are not rectangles on their published projections. But digital raster images are necessarily perfect rectangles. Clipping a digital neatline is therefore not directly comparable to cutting a paper map with scissors. Paper maps are oriented with the neatlines approximately parallel to the paper edges, but digital data sets are oriented so that the image coordinates are aligned with a selected plane ground coordinate system, or grid. Grids generally are not parallel to the quadrangle boundaries, so displayed digital data sets may appear tilted. Joining DRG quadrangles into seamless images, therefore, requires customized data organization, powerful software, or both. Storing raster images as unprojected geographic data changes the details of this problem, but does not solve it. Because digital data serve different purposes than paper maps, they can be placed on different projections. Most USGS digital products are projected on the Universal Transverse Mercator (UTM), regardless of the projection of the corresponding topographic map. UTM coordinates are unambiguous, and UTM conversions are widely implemented in software. UTM data are therefore usable by more applications, and more often, than data on any other single projection. INTRODUCTION The U.S. Geological Survey (USGS) uses a variety of projections for its maps. Projection decisions always represent tradeoffs between different types of distortion and convenience. The goals of low distortion, product standardization, and end user ease of use are not perfectly compatible. For each cartographic product, the USGS attempts to strike a reasonable balance between these conflicting goals. Of particular interest to many users are digital raster graphics (DRG). A DRG is a digital image of a paper map. The digital map is reprojected to the Universal Transverse Mercator (UTM), which creates a conflict between the data and the map collar information and in some cases visibly changes the appearance of the map image. These changes can be confusing to users. This paper explains the characteristics of Transverse Mercator projections, the relationship of plane grids to USGS map cells, and the reasons DRG’s and other USGS digital products are projected on the UTM. CYLINDRICAL PROJECTIONS There are three types of developable surfaces onto which most USGS maps are projected. They are the cylinder, the cone, and the plane (Snyder, 1987, p. 5). This paper discusses two varieties of cylindrical projections: regular and transverse cylindrical. Regular Cylindrical Projections If a cylinder is wrapped around the globe so that its surface touches the Equator (fig. 1), the meridians of longitude can be projected onto the cylinder as equally spaced straight lines perpendicular to the Equator. 1 A surface that can be transformed to a plane without distortion. The parallels of latitude are also straight lines, parallel to the equator, but not necessarily equidistant (Snyder, 1987, p. 5). The Mercator (fig. 1) is the best known regular cylindrical projection. The Mercator was designed for sea navigation because it shows rhumb lines as straight lines. Unfortunately, it is often and inappropriately used for world maps in atlases and wall charts. It presents a misleading view of the world because of excessive area distortion (Snyder and Voxland, 1989, p. 10). The Mercator shows meridians and parallels 2 The path of a ship that maintains a fixed compass direction (subject to caveats regarding the length of the line and nearness to the poles). Figure 1. Regular cylindrical projection: the Mercator. Figure 2. Transverse cylindrical projection: the Transverse Mercator. as straight lines. Regular cylindrical projections are the only commonly used projections that show both meridians and parallels as straight lines. Transverse Cylindrical Projections Setting the axis of the cylinder perpendicular to the axis of the Earth results in a transverse cylindrical projection (fig. 2). In a transverse cylindrical projection, the point of tangency between cylinder and globe is a meridian, or line of longitude, called the central meridian. Neither parallels nor meridians are straight lines, but rather are complex curves (fig. 2). The best known transverse cylindrical projection is the Transverse Mercator. The Transverse Mercator was invented by Johann Lambert (1728-77) (Snyder, 1987, p. 48), even though it is named after Gerardus Mercator (1512-94). Figures 1 and 2 illustrate that the Mercator and Transverse Mercator projections are dissimilar. On a Transverse Mercator, the central meridian (the central north-south straight line in figure 2) is the line of true scale. This makes the projection appropriate for areas with long north-south extent and narrow east-west extent. Universal Transverse Mercator In 1947 the U.S. Army adopted the UTM projection and grid for designating rectangular coordinates on large-scale maps for the entire world. The UTM is a Transverse Mercator to which specific parameters, such as standard central meridians, have been applied (Snyder, 1987, p. 57). The Earth, between latitudes 84° N. and 80° S., is divided into 60 zones, each 6° wide in longitude.