CHAPTER 4. MAP PROJECTIONS AND GRID SYSTEMS SECTION I. MAP PROJECTIONS A map projection is a method of representing a portion One common characteristic applies to all United States of the Earth’s round surface on a flat surface. Because military maps: they are all based on a conformal this procedure causes distortions of different types, projection. A conformal map projection is one that at many different projections have been developed. any point, the scale is the same in any direction and the Each projection is dictated by the size of the area angle between any two lines on the ellipsoid is the being mapped, the map scale, and the intended use of same when projected onto a plane. the maps. See table 4-2 at the end of this section. Each projection preserves certain properties and distorts others. Most projections are cylindrical, Prescribed Projections conical or azimuthal, and project an ellipsoid onto cylinders, cones or plane surfaces. These surfaces may The Transverse Mercator (TM) Projection is the be tangent to the ellipsoid or they may be secant. A preferred projection for all military mapping, though it projection is tangent to the ellipsoid when only one is not necessarily used on all military maps. The point or line of the projection surface touches the following projections are prescribed for U.S. military ellipsoid. It is secant when two points or lines touch topographic maps and charts that display a military the ellipsoid. See figure 4-1. grid on a standard scale. Military maps of non-U.S. Figure 4-1. Projection Types: Tangent and Secant. 4-2 ______________________________________________________________________________________________ MCWP 3-16.7 areas produced by other nations may not always from the point of tangency. In a projection where the conform to the following standards. U.S. maps of projected surface is secant to an ellipsoid, the scale foreign areas may be based on other projections due to factor decreases toward the central meridian or origin treaty agreements. and increases away from the points of secancy. Topographic maps at scales of 1: 500,000 or larger True ground distance can be converted to a map that lie between 80°S latitude and 84°N latitude are distance by multiplying the ground distance by the based on the TM Projection. scale factor. Topographic maps at scales of 1: 1,000,000 that lie between 80°S latitude and 84°N latitude are based on the Lambert Conformal Conic Projection. Map Scale Maps at scales of 1: 1,000,000 or larger covering the polar regions (south of 80°S latitude and north of 84°N A map scale is a representative ratio of map distances latitude) are based on the Polar Stereographic Projection. to ground distances. These ratios vary from map to map. The scale of a map is customarily chosen to Maps at scales smaller than 1: 1,000,000 are based on the correspond to the ratio at a given point or along a projection best suited for the intended use of the map. given line (if constant along that line) multiplied by a suitable scale factor (usually close to unity). It is usually expressed as a common fraction having one as a numerator and the integer closest to the actual ratio Scale Factor as a denominator. Maps used by the military vary from small-scale For most military applications, map distance and planimetric maps showing all of the continents to ground distance are considered the same. However, for large-scale topographic maps suitable for tactical some geodetic and artillery operations (especially when operations of small units and fire control. Military long distances or high accuracies are involved), it is maps are classified according to their scale. necessary to correct between map and ground distances. Small-scale: 1: 600,000 and smaller A scale factor is necessary to compensate for distortions created when projecting an ellipsoidal Medium Scale: larger than 1: 600,000; surface onto a cylinder, cone or plane depending on smaller than 1: 75,000 the projection type. The scale factor of a projection is Large Scale: 1: 75,000 and larger the ratio of arc length along a differentially small line in the plane of the projection to the arc length on the ellipsoid. This number depends on both the location of the point and on the direction of the line along which Map scales can sometimes be confusing in the sense arc length is being measured. For conformal that the scale is smaller as the number increases. This projections, the scale factor is independent of the confusion can be cleared by viewing the map scale as direction of the line and depends only on the location a fraction (1/100,000 is a smaller number than 1/ of the point. The scale factor is labeled “k”. 50,000). The following are standard scales for military maps. The scale factor is considered exact (unity) when it has a value of 1. Unity occurs at the points of tangency or 1: 1,000,000 1: 500,000 secancy between the ellipsoid and the projected 1: 250,000 1: 100,000 surface. In a projection where the projected surface is tangent to an ellipsoid, the scale factor increases away 1: 50,000 1: 25,000 Marine Artillery Survey Operations ________________________________________________________________________ 4-3 Mercator Projection The Mercator Projection is a cylindrical projection where the rotational axis of the ellipsoid coincides with the axis of the cylinder so that the Equator is tangent to the cylinder. Points on the surface of the ellipsoid are projected onto the cylinder from the origin located on the equatorial plane and vary around three-quarters of the way back from the projected area. The cylinder is then opened and flattened to produce a plane surface. The parallels of latitude and meridians of longitude both appear as sets of parallel lines that intersect at right angles. The meridians are equally spaced, but the distance between parallels increases as their distance from the Equator increases. The poles cannot be shown on this projection (the normal limits are from 80°N latitude to 80°S latitude). See figure 4-2 and figure 4-3. Figure 4-2. Mercator Projection. As the distance from the Equator increases, so does the amount of distortion; e.g., the map scale at 60°N or S latitudes is nearly twice the map scale at the Equator. Maps or charts with this projection will distort the size of an area. This is why Alaska appears to be the same size as the lower 48 states. This projection is not commonly used for military purposes except when the entire Earth must be displayed and relative positions of Figure 4-3. Mercator Projection Flattened land masses are more important than size and distance. onto a Plane. 4-4 ______________________________________________________________________________________________ MCWP 3-16.7 Transverse Mercator Projection The TM Projection is a cylindrical conformal projection. It is based on a modified Mercator Projection in that the cylinder is rotated (transverse) 90° so that the rotational axis of the ellipsoid is perpendicular to the axis of the cylinder. Generally, the TM Projection is considered as a cylinder that is secant to an ellipsoid. Only a six-degree wide portion Figure 4-4. TM Projection. of the ellipsoid is projected onto the cylinder. The centerline of the projected area is called the central meridian. The ellipsoid is then rotated six degrees inside the cylinder and another six degree portion is projected. See figure 4-4. When the TM Projection is used to project a portion of the ellipsoid onto the cylinder, the Equator and the central meridian will appear as perpendicular straight lines. A hemisphere will be distorted towards its outer edges. The shaded areas of figure 4-5 show the varying distortion of two equivalent geographic areas on the same projection. Note that both areas encompass a region 20° by 20° and are both bounded by 20° and 40°N latitude. Therefore, on the ellipsoid they are the same size. But on the projected surface the area bounded by 60° and 80° Figure 4-5. Distortion within the TM Projection. longitude is much larger than the area bounded by 0° and 20° longitude. To decrease the amount of distortion, the ellipsoid is divided into 60 6°-wide meridian onto the cylinder. A and D are the meridians projection zones, each with a meridian of longitude located three degrees from the central meridian. A’ and as its central meridian. Each zone is projected D’ are the projections of those meridians onto the between 84° 30’ N latitude and 80° 30’ S latitude. cylinder. B and C are the points where the cylinder intersects the ellipsoid creating the secant condition. The cylinder used as the projection surface for the TM Note that line BM’C is shorter than line BMC. This Projection is generally considered to be secant to the shows that any line that lies between the lines of ellipsoid as shown in figure 4-4. This means that the secancy is shorter on its projected plane (map) than it is cylinder intersects the ellipsoid in two places creating on the ellipsoid surface. Note also that lines A’B and lines of secancy that are parallel to the central meridian CD’ are longer than lines AB and CD, respectively. of the projection. The lines of secancy are located This shows that any line that lies between the lines of 180,000 meters east and west of the central meridian of secancy and the edges of the projection are longer on each projected zone. See figure 4-6.