# UNCLASSIFIED

Description of the Mercator Projection

The Mercator projection is a cylindrical, conformal projection. The equator lies on the line Y = 0. This projection is not defined at the poles. Meridians and parallels provide the framework for the Mercator projection. Meridians are projected as parallel straight lines that satisfy the equation X = a constant. Evenly spaced meridians on the ellipsoid project to evenly spaced straight lines on the projection. Parallels are projected as parallel straight lines perpendicular to meridians and satisfy the equation Y = a constant. Evenly spaced parallels on the ellipsoid project to unevenly spaced parallels on the projection. The spacing between projected parallels increases with distance from the equator. See the figure below.

A Mercator projection can be specified either in terms of a standard parallel, where the cylindrical projection surface intersects the ellipsoid and the point scale factor is 1.0, or in terms of a point scale factor at the equator.

Meridians and Parallels in the Mercator Projection

In the Mercator projection, as the latitude approaches the poles, the Y coordinate approaches infinity. Area and length distortion increases with distance from the equator. For example, the point scale factor is approximately 2 at 60° latitude and 5.7 at 80° latitude.

Description of the Transverse Mercator Projection

The Transverse Mercator projection is a transverse cylindrical, conformal projection. The line Y = 0 is the projection of the equator, and the line X = 0 is the projection of the central meridian, as shown in the figure below.

Both the central meridian and the equator are represented as straight lines. No other meridian or parallel is projected onto a straight line. Since the point scale factor is one along the central meridian, this projection is most useful near the central meridian. Scale distortion increases away from this meridian.

Meridians and Parallels in the Transverse Mercator Projection
(0 is the central meridian)

The Transverse Mercator equations for X and Y, and for latitude and longitude, are approximations. Within 4° of the central meridian, the equations for X, Y, latitude, and longitude have an error of less than 1 centimeter.

Description of the Universal Transverse Mercator (UTM) Coordinates

UTM coordinates are based on a family of projections based on the Transverse Mercator projection, in which the ellipsoid is divided into 60 longitudinal zones of 6° each. The X value, called the Easting, has a value of 500,000m at the central meridian of each zone. The Y value, called the Northing, has a value of 0m at the equator for the northern hemisphere, increasing toward the north pole, and a value of 10,000,000m at the equator for the southern hemisphere, decreasing toward the south pole. The point scale factor along the central meridian is 0.9996.

For the UTM grid system, the ellipsoid is divided into 60 longitudinal zones of 6° each. Zone number one lies between 180° E and 186° E. The zone numbers increase consecutively in the eastward direction.

Meridians and Parallels (dashed) on a UTM Grid

The area of coverage for UTM coordinates is defined by zone limits, latitude limits, and overlap.

Zone limits:

6° zones, extending 3° to each side of the central meridian.

Zone overlap:

40 km on either side of the zone limits.

Latitude limits:

North: 84° N

South: 80°N

Polar overlap:

30' toward the poles

North: 84° 30'N

South: 80° 30'S

Description of the Polar Stereographic Projection

The Polar Stereographic projection is an azimuthal projection. It is the limiting case of the Lambert Conformal Conic projection when the standard parallels approach one of the poles. In this conformal projection meridians are straight lines, and parallels are concentric circles. The Central Meridian parameter determines the orientation of the projection. A value of zero results in projections as shown in the figure below. Increasing this value rotates the north polar projection clockwise, and the south polar projection counterclockwise.

A Polar Stereographic projection can be specified either in terms of a standard parallel, where the planar projection surface intersects the ellipsoid and the point scale factor is 1.0, or in terms of a point scale factor at the pole.

Meridians and Parallels in the Polar Stereographic Projection

Description of the Universal Polar Stereographic (UPS) Coordinates

Universal Polar Stereographic (UPS) is the standard military grid used in polar regions. UPS is a family of two projections based on the Polar Stereographic projection, one for each of the poles. Both the X value, called the Easting, and the Y value, called the Northing, have values of 2,000,000m at the poles. The point scale factor at each pole is 0.9994.

Meridians and Parallels on a UPS Grid
(North zone)

Zone limits:

North zone: 84°N to 90°N

South zone: 80°S to 90°S

UTM overlap: 30' overlap

North: 83° 30'N

South: 79° 30'S

Description of Albers Equal Area Conic Projection

The Albers Equal Area Conic projection is a conical, equal area projection. As shown in the figure below, the meridians are equally spaced, straight, converging lines. The angles between the meridians are less than the true angles. Meridians intersect the parallels at right angles. Parallels are unequally spaced arcs of concentric circles. The parallels are closer together at the northernmost and southernmost regions of the map. They are further apart in the latitudes between the standard parallels. The poles are normally circular arcs that enclose the same angle as that enclosed by the other parallels for a given range of longitude. The Albers Equal Area Conic projection is symmetrical about any meridian.

Albers Equal Area Conic Projection
(Origin Latitude = 45°N, Standard Parallels = 40°N & 50°N)

Scale is true along the two standard parallels. It is also true even when there is only one standard parallel. Standard parallels should be chosen to minimize scale variations. Scale is true along any given parallel. The scale factor along the meridians is the reciprocal of the scale factor along the parallels, to retain equal area.

The Albers Equal Area Conic projection is free of scale and shape distortion along either the one or two standard parallels. Along any given parallel, distortion is constant.

The standard parallels can not both be 0° or the opposite sign of each other, as this would cause the cone to become a cylinder.

Description of Azimuthal Equidistant Projection

The Azimuthal Equidistant projection is an azimuthal, equidistant, non-perspective projection. As shown in the figure below, the meridians are straight lines on the polar aspect and complex curves on the equatorial and oblique aspects. The central meridian on the equatorial and oblique aspects is a straight line. Parallels on the polar aspect are circles, equally spaced, centered at the pole, which is a point. Parallels on the equatorial and oblique aspects are complex curves equally spaced along the central meridian. The equator is a straight line on the equatorial aspect. The projection is symmetrical about any meridian for the polar aspect, the equator or central meridian for the equatorial aspect, and the central meridian for the oblique aspect.

Azimuthal Equidistant Projection

Scale is true along any straight line radiating from the center of the projection. A point opposite the center is projected as a circle twice the radius of the mapped equator. Scale along this circle is infinite.

The projection is free of distortion at the center. Distortion is severe for a world map.

Description of Bonne Projection

The Bonne projection is a pseudoconical, equal area projection. As shown in the figure below, the central meridian is a straight line, while other meridians are complex curves which connect equally spaced points along each parallel of latitude and concave toward the central meridian. Parallels are concentric circular arcs spaced at true distances along the central meridian. The curvature of the standard parallel is equal to that of its curvature on a cone tangent at that latitude. The poles are points.

Bonne Projection
(Origin Latitude = 45°N)

Scale is true along the central meridian and each parallel.

There is no distortion along the central meridian and the standard parallel. As distance from the central meridian and the standard parallel increases, shape distortion increases and meridians do not intersect parallels at right angles.

Sinusoidal is a limiting form of the Bonne projection with the standard parallel at the equator. The equations must be rewritten, since the parallels of latitude are straight.

Description of British National Grid Coordinates

The British National Grid Reference System is an alphanumeric system, based on the Transverse Mercator map projection, for identifying positions. A British National Grid coordinate consists of an alphabetic 500,000 unit grid square identifier, an alphabetic 100,000 unit grid square identifier, and grid coordinates expressed to a given precision.

British National Grid

British National Grid parameters are fixed at an Origin Latitude of 49°N, Central Meridian of 2°W, False Easting of 400,000 meters, False Northing of -100,000 meters and a Scale Factor of .9996012717.

British National Grid uses only the Airy ellipsoid.

The coordinate SV 0000000000 is located at an Easting of 0m and a Northing of 0m in Transverse Mercator coordinates, which is the bottom, left most coordinate in the figure above.

Description of Cassini Projection

The Cassini projection is a cylindrical, equidistant projection. As shown in the figure below, the central meridian, each meridian 90° from the central meridian, and the equator are straight lines. Other meridians and parallels are complex curves, which are concave toward the central meridian and the nearest pole. The poles are points along the central meridian. Cassini is symmetrical about any straight meridian or the equator.

Cassini Projection

Scale is true along the central meridian and lines perpendicular to the central meridian. Scale increases with distance from the central meridian, along a direction parallel to the central meridian.

There is no distortion along the central meridian. If the longitude is greater than 4° from the central meridian, distortion will result. Horizontal straight lines, near the upper and lower limits, represent microscopic circles on the globe 90° from the central meridian.

Description of Cylindrical Equal Area Projection

The Cylindrical Equal Area projection is a cylindrical, equal area projection. It is an orthographic projection of a sphere onto a cylinder. As shown in the figure below, the meridians are equally spaced, straight, parallel lines almost 1/3 the length of the equator. The parallels are unequally spaced, straight parallel lines perpendicular to the meridians. The parallels are spaced in proportion to the sine of the latitude from the equator. The poles are straight lines as long as the equator. The Cylindrical Equal Area projection is symmetrical about the equator or any meridian.

Cylindrical Equal Area Projection

Scale is true along the equator. In the direction of the parallels, scale increases with distance from the equator and decreases in the direction of the meridians. Parallels of opposite sign have the same scale.

The Cylindrical Equal Area projection does not have area distortion anywhere. There is severe shape distortion at the poles.

Description of Eckert IV Projection

The Eckert IV projection is a pseudocylindrical, equal area projection. As shown in the figure below, the central meridian is a straight line half as long as the equator. The 180° east and west meridians are semicircles. All other meridians are equally spaced elliptical arcs. The parallels are unequally spaced, straight, parallel lines that are farthest apart at the equator. The parallels are perpendicular to the central meridian. The poles are straight lines half as long as the equator. Eckert IV is symmetrical about the central meridian or the equator.

Eckert IV Projection

Scale is true along latitudes 40°30´ N. and S. For any given latitude and the latitude of opposite sign, scale is constant.

Eckert IV is free of distortion only at latitudes 40°30´ N. and S. at the central meridian. The Eckert IV projection is used only in the spherical form.

Description of Eckert VI Projection

The Eckert VI projection is a pseudocylindrical, equal area projection. As shown in the figure below, the central meridian is a straight line half as long as the equator. The other meridians are equally spaced sinusoidal curves. The parallels are unequally spaced, straight, parallel lines that are farthest apart at the equator. The parallels are perpendicular to the central meridian. The poles are straight lines half as long as the equator. Eckert VI is symmetrical about the central meridian or the equator.

Eckert VI Projection

Scale is true along latitudes 49°16´ N. and S. For any given latitude and the latitude of opposite sign, scale is constant.

Eckert VI is free of distortion only at latitudes 49°16´ N. and S. at the central meridian. The Eckert VI projection is used only in the spherical form.

Description of Equidistant Cylindrical Projection

The Equidistant Cylindrical projection is a cylindrical equidistant projection. As shown in the figure below, the meridians are equally spaced, straight, parallel lines more than half as long as the equator. The parallels are equally spaced, straight, parallel lines perpendicular to the meridians. Meridian spacing is four-fifths of the parallel spacing. The poles are straight lines as long as the equator. The Equidistant Cylindrical projection is symmetrical about any meridian or the equator.

Equidistant Cylindrical Projection

Scale is true along two standard parallels equidistant from the equator and along all meridians. Scale is small along the equator but increases along the parallels with distance from the equator. For any given parallel, scale is constant and equal to the scale at the parallel of opposite sign.

Infinitesimally small circles on the globe are circles on the map at the chosen standard parallels of 30° N. and S. Area and local shape are distorted everywhere else. The Equidistant Cylindrical projection is used only in the spherical form.

Description of Gnomonic Projection

The Gnomonic projection is an azimuthal, perspective projection. It is neither conformal nor equal area. As shown in the figure below, he equator and all meridians are straight lines. For the polar aspect, meridians are equally spaced and intersect at the pole. Meridians are unequally spaced for the oblique and equatorial aspects. Except for the equator and the poles, all parallels are circles, parabolas or hyperbolas. The pole is a point on the polar aspect. On the equatorial aspect, poles cannot be shown.

Gnomonic Projection

Scale is true only where the central line crosses the central meridian. It rapidly increases with distance from the center of the projection.

The projection is free of distortion only at the center. It rapidly increases with distance from the center of the projection.

The Gnomonic projection is used only in the spherical form.

Description of Lambert Conformal Conic Projection

The Lambert Conformal Conic projection is a conformal projection in which the projected parallels are unequally spaced arcs of concentric circles centered at the pole, as shown in the figure below. Spacing of parallels increases away from the origin latitude. The projected meridians are equally spaced radii of concentric circles that meet at the pole.

A Lambert Conformal Conic projection can be specified using either one or two standard parallels. In the case where there is one standard parallel, the point scale factor along that parallel is also specified. In the case where there are two standard parallels, the point scale factor is one along both of those standard parallels, and is less than one in the area between them. The point scale factor increases as a point moves outward from the standard parallel(s). The two standard parallels are generally placed at one-sixth and five-sixths of the range of latitudes to be included. When the two standard parallels are both set to the same latitude value, the result is a Lambert Conformal Conic projection with one standard parallel at the specified latitude.

If there are two standard parallels that are symmetrical about the equator, the Mercator projection results. If there is only one standard parallel and it is at a pole, the Polar Stereographic projection results.

The standard parallels cannot both be 0° or the opposite sign of each other, as this would cause the cone to become a cylinder.

The pole closest to a standard parallel is a point while the other pole is at infinity. Lambert is symmetrical about any meridian.

Lambert Conformal Conic Projection

The scale is constant along any given parallel and is the same in all directions at a given point.

Lambert is free of distortion only along the standard parallel(s). Distortion is constant along any given parallel and conformal everywhere but at the poles.

Description of Miller Cylindrical Projection

The Miller Cylindrical projection is a cylindrical projection that is neither conformal nor equal area. As shown in the figure below, the meridians are equally spaced, straight, parallel lines 73% as long as the equator. The parallels are unequally spaced, straight lines perpendicular to the meridians. Parallel spacing increases with distance from the equator. The poles are straight lines the same length as the equator. The Miller Cylindrical projection is symmetrical about any meridian or the equator.

Miller Cylindrical Projection

In all directions along the equator, scale is true. At any other given latitude, scale is constant in any given direction. Latitudes of opposite sign have the same scale. Scale changes with latitude and direction.

The projection is free of distortion at the equator. Shape, area and scale distortion increase slightly away from the equator. At the poles, distortion becomes severe.

The Miller Cylindrical projection is used only in the spherical form.

Description of Mollweide Projection

The Mollweide projection is a pseudocylindrical, equal area projection. As shown in the figure below, the central meridian is a straight line half as long as the equator. The 90° east and west meridians are circular arcs. All other meridians are equally spaced, elliptical arcs. The parallels are unequally spaced, straight, parallel lines perpendicular to the central meridian. The parallels are farthest apart near the equator. The poles are points. The Mollweide projection is symmetrical about the central meridian or the equator.

Mollweide Projection

Scale is true along latitudes 40°44´ N. and S. Scale is the same for latitudes of the opposite sign and is constant along any given latitude.

Distortion is severe near the outer meridians at high latitudes. The projection is free of distortion only at latitudes 40°44´ N. and S. on the central meridian.

The Mollweide projection is used only in the spherical form.

Description of New Zealand Map Grid Projection

The New Zealand Map Grid projection is conformal, but otherwise is unlike any other mapping projection. The projection gives a small range of scale variation over New Zealand, which lies between 166° and 180° East longitude and 34° and 48° South latitude. Meridians and parallels are lines. The central meridian, which is not straight, is oriented so that its tangent at the origin is the north-south axis of coordinates.

New Zealand Map Grid Projection

New Zealand Map Grid parameters are fixed at an Origin Latitude of 41°S, Central Meridian of 173°E, False Easting of 2,510,000 meters and a False Northing of 6,023,150 meters.

It uses only the International ellipsoid and the Geodetic Datum 1949 datum.

Easting values are always less than 5,000,000 meters and Northing values are always greater than 5,000,000 meters. Easting values for the land area of New Zealand range from 2,000,000 to 3,000,000 meters and Northing values range from 5,300,000 to 6,800,000 meters.

Description of Ney's (Modified Lambert Conformal Conic) Projection

The Ney's (Modified Lambert Conformal Conic) projection is a conformal projection in which the projected parallels are expanded slightly to form complete concentric circles centered at the pole. The projected meridians are radii of concentric circles that meet at the pole. Ney's is a limiting form of the Lambert Conformal Conic. There are two parallels, called standard parallels, along which the point scale factor is one. The first standard parallel is at either ±71 or ±74 degrees. The second standard parallel is at ±89 59 58.0 degrees, in the same hemisphere as the first standard parallel.

Ney's (Modified Lambert Conformal Conic) Projection
(Origin Latitude = 80°N, Standard Parallels = 71°N & 89 59 58.0°N)

Ney's (Modified Lambert Conformal Conic) is used near the poles. Scale distortion is small 25° to 30° from the pole. Distortion rapidly increases beyond this.

Description of Oblique Mercator Projection

The Oblique Mercator projection is an oblique, cylindrical, conformal projection. As shown in the figure below, there are two meridians which are straight lines 180° apart. Other meridians and parallels are complex curves. The poles are points that do not lie on the central line. The projection is symmetrical about any straight meridian.

Oblique Mercator Projection

On the spherical aspect, scale is true along the central line, a great circle at an oblique angle, or along two straight lines parallel to the central line. Scale is constant along any straight line parallel to the central line. It becomes infinite 90° from the central line. Scale on the ellipsoidal aspect is similar, but varies slightly.

Distortion is the same as that of the Mercator projection, at a given distance.

Mercator is a limiting form of Oblique Mercator with the equator as the central line. The Transverse Mercator projection is a limiting form of Oblique Mercator with a meridian as the central line.

Description of Orthographic Projection

The Orthographic projection is an azimuthal, perspective projection that is neither conformal nor equal area. Only one hemisphere can be shown at a time, as shown in the figure below.

The central meridian in the equatorial aspect is a straight line. The 90° meridians form a circle representing the limit of the equatorial aspect. Other meridians are unequally spaced, ellipses of eccentricities ranging from 0 (the bounding circle) to 1.0 (the central meridian). Meridian spacing decreases away from the central meridian. Parallels are unequally spaced, straight, parallel lines perpendicular to the central meridian. Parallel spacing decreases away from the equator. Parallels intersect the outer meridian at equal intervals. The projection is symmetrical about the central meridian or equator.

Orthographic Projection (Equatorial Aspect)

For the polar aspect, meridians are equally spaced straight lines radiating from the pole at their true angles. Parallels are unequally spaced circles centered at the pole. The pole is a point. Parallel spacing decreases away from the pole. The projection is symmetrical about any meridian.

Orthographic Projection (North Polar Aspect)

The central meridian in the oblique aspect is also a straight line. Other meridians are ellipses of varying eccentricities. Meridian spacing decreases away from the central meridian. Parallels are complete or partial ellipses with the same eccentricity, whose minor axes lie along the central meridian. Parallel spacing decreases from the center of the projection.

Orthographic Projection (Oblique Aspect)

Scale is true at the center of the projection and along all circles drawn about the center. The scale is true only in the direction of the circumference and it decreases radially with distance from the center.

The center of the projection is free of distortion. Distortion quickly increases with distance from the center. At the outer regions, distortion is severe.

The Orthographic projection is used only in the spherical form.

Description of Polyconic Projection

The Polyconic projection is neither conformal nor equal area. As shown in the figure below, the central meridian is a straight line, while all other meridians are complex curves equally spaced along the equator and each parallel. The equator is a straight line, while all other parallels are nonconcentric, circular arcs spaced at true intervals along the central meridian. Each parallel has a curvature developed from a cone tangent at that latitude. The poles are points. The Polyconic projection is symmetrical about the central meridian and the equator.

Polyconic Projection

Scale is true along the central meridian and each parallel. No parallel is standard in that it has correct angles, except at the central meridian, because the meridians are lengthened by different amounts to cross each parallel at the correct position along the parallel.

The Polyconic projection is free of distortion only along the central meridian. If the range extends east or west a great distance, a large amount of distortion will result.

Description of Sinusoidal Projection

The Sinusoidal projection is a pseudocylindrical, equal area projection. The central meridian is a straight line half as long as the equator. All other meridians are equally spaced sinusoidal curves that intersect at the poles. The parallels are equally spaced, straight, parallel lines perpendicular to the meridians. The poles are shown as points. The Sinusoidal projection is symmetrical about the central meridian or the equator.

Sinusoidal Projection

Scale is true along the central meridian and every parallel.

The Sinusoidal projection is free of distortion along the central meridian and equator. At high latitudes near the outer meridians, especially in the polar regions, distortion is extreme. An interrupted form of the projection involving several central meridians helps reduce distortion.

Description of Stereographic Projection

The Stereographic projection is an azimuthal, conformal, true perspective (for the sphere) projection in which meridians are straight lines on the polar aspect and arcs of circles on the oblique and equatorial aspects. For all aspects, the central meridian is a straight line. Parallels are concentric circles, except for the equator on the equatorial aspect. It is a straight line. On the oblique aspect, the parallel opposite in sign to the origin latitude is also a straight line. For the polar aspect, the opposite pole cannot be shown.

Stereographic Projection

Scale is true at the intersection of the origin latitude and central meridian. Scale is constant along any circle whose center is at the center of the projection. Scale increases away from the projection center.

The projection is free of distortion at the center.

Description of the Transverse Cylindrical Equal Area Projection

The Transverse Cylindrical Equal Area projection is a transverse aspect of the normal Cylindrical Equal Area projection. It is a perspective projection onto a cylinder tangent or secant at an oblique angle, or centered on a meridian. In the transverse aspect, the central meridian, each meridian 90° from the central meridian and the equator are straight lines. All other meridians and parallels are complex curves. The poles are straight lines.

Transverse Cylindrical Equal Area Projection

Scale is true along the central meridian, or along two approximately (for the ellipsoid) straight lines equidistant from and parallel to the central meridian.

There is not any distortion of area. There is no scale and shape distortion at the standard parallel, but there is severe scale and shape distortion 90° from the central meridian.

Description of Van der Grinten Projection

The Van Der Grinten projection is neither equal area nor conformal and it is not pseudocylindrical. It shows the entire globe enclosed in a circle. The central meridian is a straight line and the other meridians are arcs of circles equally spaced along the equator. The equator is a straight line and the other parallels are arcs of circles. Parallel spacing increases with latitude. The 75th parallels are shown to be halfway between the equator and the poles. The poles are shown as points. The Van Der Grinten projection is symmetrical along the central meridian or equator.

Van der Grinten Projection

Scale is true along the equator. It quickly increases with distance from the equator.

There is a large amount of area distortion near the poles.

The Van Der Grinten projection is used only in the spherical form.

# UNCLASSIFIED