Description of the Mercator Projection
The Mercator projection is a conformal projection for which the point scale factor is one along the equator. 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.
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. GEOTRANS limits latitude values to between 89.5°S and 89.5°N.
Description of the Transverse Mercator Projection
The Transverse Mercator projection is a conformal projection for which the point scale factor equals one along the central meridian. 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. It is intended that a UTM projection should be used only in one of the 6-degree zones, plus the overlap area.
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. Note: there are several exceptions to this general rule.
Latitude limits:
North: 84° N
South: 80°N
Zone overlap:
40 km on either side of the zone limits.
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 a 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 Longitude Down from Pole 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.
Meridians and Parallels in the Polar Stereographic Projection
The Latitude of True Scale parameter determines the latitude at which the scale factor is one.
There is no general agreement on the area of coverage for the Polar Stereographic projection. GEOTRANS allows each Polar Stereographic projection to extend as far as the equator.
The Easting\X and Northing\Y coordinates range from -12,713,601.0 to 12,713,601.0 at an origin latitude of 90° and a central meridian of 0°. These ranges change as the central meridian and origin latitude are varied. They can be found at a latitude of 0° and a longitude equal to the central meridian.
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)
The limits of the system are north of 84°N and south of 80°S. In order to provide a 30-minute overlap with the UTM grid, the UPS grid contains an overlap that extends to 83° 30'N and 79° 30'S. The area of coverage for UPS coordinates is defined by zone limits and overlap.
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 Lambert Equal Area Conic projection is a limiting form of Albers if the pole is chosen as one of the two standard parallels. If the pole is the only standard parallel, the Lambert Azimuthal Equal Area projection results. If the equator is the only standard parallel, the Lambert Cylindrical Equal Area projection results, but the formulas need to be modified.
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.
The Easting\X and Northing\Y coordinates range from -40,000,000 to 40,000,000.
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.
The Easting\X and Northing\Y coordinates range from -19,903,915.0 to 19,903,915.0 at a central meridian of 0° and an origin latitude of 0°. These ranges change as the central meridian and origin latitude are varied. Northing\Y ranges from ±19,903,915.0 to ±20,015,110.0. The range of Easting\X values can be found at latitudes of ± origin latitude and at longitudes of central meridian ± PI - 1 degree. These ranges depend upon the direction of the central meridian.
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.
The Easting\X and Northing\Y coordinates range from -40,000,000 to 40,000,000.
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,
British National Grid uses only the Airy ellipsoid.
GEOTRANS limits the latitude values to between 49.5°N and 61.5°N. Longitude values are limited to between 10°W and 3.5°E. Easting values range from -133134.0 to 759961.0 and Northing values range from -14829.0 to 1257875.0.
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.
The Easting\X coordinate ranges from -20,037,508.4 to 20,037,508.4 and the Northing\Y coordinate ranges from -56,575,846.0 to 56,575,846.0 at a central meridian of 0° and an origin latitude of 0°. These ranges change as the central meridian and origin latitude are varied. Easting\X ranges from ±19,926,188.9 to ±20,037,508.4. The range of Northing\Y values can be found at a latitude of ± 30° and a longitude of PI ± central meridian. These ranges depend upon the direction of 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.
The Easting\X coordinate ranges from -20,037,509.0 to 20,037,509.0 and the Northing\Y coordinate ranges from -6,363,886.0 to 6,363,886.0 at a central meridian of 0° and an origin latitude of 0°. These ranges change as the central meridian and origin latitude are varied. The range of Easting\X values can be found at a latitude of PI /2 and a longitude of central meridian ± PI ± 1 and central meridian ± PI , depending on the direction of the central meridian. The range of Northing\Y values can be found at a latitude of PI /2 and a longitude of PI .
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.
The Easting\X coordinate ranges from -16,902,288.0 to 16,902,288.0 and the Northing\Y coordinate ranges from -8,451,144.0 to 8,451,144.0 at a central meridian of 0° and an origin latitude of 0°. As the central meridian and origin latitude are varied, the Easting\X coordinate ranges from ± 16,808,386.0 to ± 16,902,288.0, depending on the direction of the central meridian.
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.
The Easting\X coordinate ranges from -17,653,838.0 to 17,653,838.0 and the Northing\Y coordinate ranges from -8,826,919.0 to 8,826,919.0 at a central meridian of 0° and an origin latitude of 0°. As the central meridian and origin latitude are varied, the Easting\X coordinate ranges from ± 17,555,761.0 to ± 17,653,838.0, depending on the direction of the central meridian.
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.
The Easting\X coordinate ranges from -20,015,110.0 to 20,015,110.0 and the Northing\Y coordinate ranges from -10,007,555.0 to 10,007,555.0 at a central meridian of 0° and a standard parallel of 0°. The range of the Easting\X coordinate changes as the central meridian and standard parallel are varied. The range of Easting\X values can be found at a latitude of PI /2 and a longitude of central meridian ± PI ± 1 and central meridian ± PI , depending on the direction of the central meridian.
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.
The Easting\X and Northing\Y coordinates range from -40,000,000.0 to 40,000,000.0.
Description of Lambert Conformal Conic (1 parallel) Projection
The Lambert Conformal Conic (1 parallel) 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.
In this case, there is one parallel, called the standard parallel, along which there exists a point scale factor. 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 (1 parallel) Projection
(Origin Latitude = 45°N, Scale Factor = 1.0)
The family of Lambert conformal conic projections and their limiting cases is used worldwide. For a given member of this family, the point scale factor increases as a point moves away from the standard parallel. 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. Distortion is constant along any given parallel and conformal everywhere but at the poles.
If the standard parallel is at the equator, the Mercator projection results. If the standard parallel is at a pole, the Polar Stereographic projection results.
The standard parallel cannot be 0°. GEOTRANS limits the standard parallel values to between 89 59 59.0°S and 89 59 59.0°N.
The Easting\X and Northing\Y coordinates range from -40,000,000 to 40,000,000.
Description of Lambert Conformal Conic (2 parallel) Projection
The Lambert Conformal Conic (2 parallel) 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.
In this case, there are two parallels, called standard parallels, along which the point scale factor is one. The standard parallels are generally placed one-sixth and five-sixths of the range of latitudes. 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 (2 parallel) Projection
(Origin Latitude = 45°N, Standard Parallels = 40°N & 50°N)
The family of Lambert conformal conic projections and their limiting cases is used worldwide. For a given member of this family, the point scale factor increases as a point moves away from the standard parallel(s). 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 parallels. Distortion is constant along any given parallel and conformal everywhere but at the poles.
When
the two standard parallel parameters 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 the standard parallels 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. GEOTRANS limits the standard parallel values to between 89 59 59.0°S and 89 59 59.0°N.
The Easting\X and Northing\Y
coordinates range from -40,000,000 to 40,000,000.
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.
The Easting\X coordinate ranges from -20,015,110.0 to 20,015,110.0 and the Northing\Y coordinate ranges from -14,675,058.0 to 14,675,058.0 at a central meridian of 0° and an origin latitude of 0°. As the central meridian and origin latitude are varied, the Easting\X coordinate ranges from ± 19,903,915.0 to ± 20,015,110.0, depending on the direction of the central meridian.
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.
The Easting\X coordinate ranges from -18,019,930.0 to 18,019,930.0 and the Northing\Y coordinate ranges from -9,009,965.0 to 9,009,965.0 at a central meridian of 0° and an origin latitude of 0°. As the central meridian and origin latitude are varied, the Easting\X coordinate ranges from ± 17,919,819.0 to ± 18,019,930.0, depending on the direction of the central meridian.
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. One parallel is at either ±71 or ±74 degrees. The other parallel is at ±89 59 59.0 degrees, depending on which hemisphere the first parallel is in.
Ney's (Modified
Lambert Conformal Conic) Projection
(Origin Latitude = 80°N, Standard Parallels = 71°N & 89 59 59.0°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.
The Easting\X and Northing\Y coordinates range from -40,000,000 to 40,000,000.
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.
The Easting\X and Northing\Y coordinates range from -40,000,000 to 40,000,000.
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.
The Easting\X and Northing\Y coordinates range from -6,371,007.1810824 to 6,371,007.1810824.
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.
The Easting\X coordinate ranges from -20,037,509.0 to 20,037,509.0 and the Northing\Y coordinate ranges from -15,348,215.0 to 15,348,215.0 at a central meridian of 0° and an origin latitude of 0°. These ranges change as the central meridian and origin latitude are varied. Easting\X ranges from ±19,926,189.0 to ±20,037,509.0. The range of Northing\Y values can be found at a latitude of ± 41° and a longitude of PI ± central meridian. These ranges depend upon the direction of the central meridian.
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.
The Easting\X coordinate ranges from -20,037,509.0 to 20,037,509.0 and the Northing\Y coordinate ranges from -10,001,966.0 to 10,001,966.0 at a central meridian of 0° and an origin latitude of 0°. As the central meridian and origin latitude are varied, the Easting\X coordinate ranges from ± 19,926,189.0 to ± 20,037,509.0, depending on the direction of the central meridian.
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.
The Easting\X and Northing\Y coordinates range from -1,460,090,226.0 to 1,460,090,226.0 at a central meridian of 0° and an origin latitude of 0°. These ranges change as the central meridian and origin latitude are varied. The range of Easting\X values can be found at latitudes of -origin latitude and at longitudes of central meridian ± PI - 1 degree. These ranges depend upon the direction of the central meridian.
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.
The Easting\X coordinate ranges from -6,398,628.0 to 6,398,628.0 and the Northing\Y coordinate ranges from -20,003,931.0 to 20,003,931.0 at a central meridian of 0° and an origin latitude of 0°. The range of the Northing\Y coordinate changes as the central meridian and origin latitude are varied. The range of Northing\Y values can be found at a latitude of PI /2 and a longitude of PI . This value then needs to be added to ±20,003,931.458986, depending on the direction of 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.
The Easting\X and Northing\Y coordinates range from -20,015,109.356056 to 20,015,109.356056.