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(Those coefficients can be improved, but as they stand the distance they give is correct within a centimeter.)
The formulae both return units of meters per degree.
An alternative method to estimate the length of a longitudinal degree at latitude is to assume a spherical Earth (to get the width per minute and second, divide by 60 and 3600, respectively):
where Earth's average meridional radius is 6,367,449 m. Since the Earth is an oblate spheroid, not spherical, that result can be off by several tenths of a percent; a better approximation of a longitudinal degree at latitude is
where Earth's equatorial radius equals 6,378,137 m and ; for the GRS80 and WGS84 spheroids, b/a calculates to be 0.99664719. ( is known as the reduced (or parametric) latitude). Aside from rounding, this is the exact distance along a parallel of latitude; getting the distance along the shortest route will be more work, but those two distances are always within 0.6 meter of each other if the two points are one degree of longitude apart.
Latitude  City  Degree  Minute  Second  ±0.0001° 

60°  Saint Petersburg  55.80 km  0.930 km  15.50 m  5.58 m 
51° 28′ 38″ N  Greenwich  69.47 km  1.158 km  19.30 m  6.95 m 
45°  Bordeaux  78.85 km  1.31 km  21.90 m  7.89 m 
30°  New Orleans  96.49 km  1.61 km  26.80 m  9.65 m 
0°  Quito  111.3 km  1.855 km  30.92 m  11.13 m 
To establish the position of a geographic location on a map, a map projection is used to convert geodetic coordinates to plane coordinates on a map; it projects the datum ellipsoidal coordinates and height onto a flat surface of a map. The datum, along with a map projection applied to a grid of reference locations, establishes a grid system for plotting locations. Common map projections in current use include the Universal Transverse Mercator (UTM), the Military Grid Reference System (MGRS), the United States National Grid (USNG), the Global Area Reference System (GARS) and the World Geographic Reference System (GEOREF).^{ [11] } Coordinates on a map are usually in terms northing N and easting E offsets relative to a specified origin.
Map projection formulas depend on the geometry of the projection as well as parameters dependent on the particular location at which the map is projected. The set of parameters can vary based on the type of project and the conventions chosen for the projection. For the transverse Mercator projection used in UTM, the parameters associated are the latitude and longitude of the natural origin, the false northing and false easting, and an overall scale factor.^{ [12] } Given the parameters associated with particular location or grin, the projection formulas for the transverse Mercator are a complex mix of algebraic and trigonometric functions.^{ [12] }^{: 4554 }
The Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS) coordinate systems both use a metricbased Cartesian grid laid out on a conformally projected surface to locate positions on the surface of the Earth. The UTM system is not a single map projection but a series of sixty, each covering 6degree bands of longitude. The UPS system is used for the polar regions, which are not covered by the UTM system.
During medieval times, the stereographic coordinate system was used for navigation purposes.^{[ citation needed ]} The stereographic coordinate system was superseded by the latitudelongitude system. Although no longer used in navigation, the stereographic coordinate system is still used in modern times to describe crystallographic orientations in the fields of crystallography, mineralogy and materials science.^{[ citation needed ]}
This section needs expansion. You can help by adding to it. (December 2018) 
Vertical coordinates include height and depth.
Every point that is expressed in ellipsoidal coordinates can be expressed as an rectilinear x y z (Cartesian) coordinate. Cartesian coordinates simplify many mathematical calculations. The Cartesian systems of different datums are not equivalent.^{ [2] }
The Earthcentered Earthfixed (also known as the ECEF, ECF, or conventional terrestrial coordinate system) rotates with the Earth and has its origin at the center of the Earth.
The conventional righthanded coordinate system puts:
An example is the NGS data for a brass disk near Donner Summit, in California. Given the dimensions of the ellipsoid, the conversion from lat/lon/heightaboveellipsoid coordinates to XYZ is straightforward—calculate the XYZ for the given latlon on the surface of the ellipsoid and add the XYZ vector that is perpendicular to the ellipsoid there and has length equal to the point's height above the ellipsoid. The reverse conversion is harder: given XYZ we can immediately get longitude, but no closed formula for latitude and height exists. See "Geodetic system." Using Bowring's formula in 1976 Survey Review the first iteration gives latitude correct within 10^{11} degree as long as the point is within 10,000 meters above or 5,000 meters below the ellipsoid.
A local tangent plane can be defined based on the vertical and horizontal dimensions. The vertical coordinate can point either up or down. There are two kinds of conventions for the frames:
In many targeting and tracking applications the local ENU Cartesian coordinate system is far more intuitive and practical than ECEF or geodetic coordinates. The local ENU coordinates are formed from a plane tangent to the Earth's surface fixed to a specific location and hence it is sometimes known as a local tangent or local geodetic plane. By convention the east axis is labeled , the north and the up .
In an airplane, most objects of interest are below the aircraft, so it is sensible to define down as a positive number. The NED coordinates allow this as an alternative to the ENU. By convention, the north axis is labeled , the east and the down . To avoid confusion between and , etc. in this article we will restrict the local coordinate frame to ENU.
In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the geodetic latitude as defined below. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six auxiliary latitudes that are used in special applications.
The Mercator projection is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because it is unique in representing north as up and south as down everywhere while preserving local directions and shapes. The map is thereby conformal. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation is very small near the equator but accelerates with increasing latitude to become infinite at the poles. As a result, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator, such as Central Africa.
In mathematics, a spherical coordinate system is a coordinate system for threedimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the threedimensional version of the polar coordinate system.
An azimuth is an angular measurement in a spherical coordinate system. The vector from an observer (origin) to a point of interest is projected perpendicularly onto a reference plane; the angle between the projected vector and a reference vector on the reference plane is called the azimuth.
Earth radius is the distance from the center of Earth to a point on or near its surface. Approximating the figure of Earth by an Earth spheroid, the radius ranges from a maximum of nearly 6,378 km (3,963 mi) to a minimum of nearly 6,357 km (3,950 mi).
In navigation, a rhumb line, rhumb, or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true north.
In geodesy, conversion among different geographic coordinate systems is made necessary by the different geographic coordinate systems in use across the world and over time. Coordinate conversion is composed of a number of different types of conversion: format change of geographic coordinates, conversion of coordinate systems, or transformation to different geodetic datums. Geographic coordinate conversion has applications in cartography, surveying, navigation and geographic information systems.
In geodesy, a reference ellipsoid is a mathematically defined surface that approximates the geoid, which is the truer, imperfect figure of the Earth, or other planetary body, as opposed to a perfect, smooth, and unaltered sphere, which factors in the undulations of the bodies' gravity due to variations in the composition and density of the interior, as well as the subsequent flattening caused by the centrifugal force from the rotation of these massive objects . Because of their relative simplicity, reference ellipsoids are used as a preferred surface on which geodetic network computations are performed and point coordinates such as latitude, longitude, and elevation are defined.
The transverse Mercator map projection is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the Universal Transverse Mercator. When paired with a suitable geodetic datum, the transverse Mercator delivers high accuracy in zones less than a few degrees in eastwest extent.
A geodetic datum or geodetic system is a global datum reference or reference frame for precisely measuring locations on Earth or other planetary bodies. Datums are crucial to any technology or technique based on spatial location, including geodesy, navigation, surveying, geographic information systems, remote sensing, and cartography. A horizontal datum is used to measure a location across the Earth's surface, in latitude and longitude or another coordinate system; a vertical datum is used to measure the elevation or depth relative to a standard origin, such as mean sea level (MSL). Since the rise of the global positioning system (GPS), the ellipsoid and datum WGS 84 it uses has supplanted most others in many applications. The WGS 84 is intended for global use, unlike most earlier datums.
The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways.
The equirectangular projection (also called the equidistant cylindrical projection or la carte parallélogrammatique projection, and which includes the special case of the plate carrée projection, is a simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100. The projection maps meridians to vertical straight lines of constant spacing, and circles of latitude to horizontal straight lines of constant spacing. The projection is neither equal area nor conformal. Because of the distortions introduced by this projection, it has little use in navigation or cadastral mapping and finds its main use in thematic mapping. In particular, the plate carrée has become a standard for global raster datasets, such as Celestia, NASA World Wind, and Natural Earth, because of the particularly simple relationship between the position of an image pixel on the map and its corresponding geographic location on Earth. In addition it is frequently used in panoramic photography to represent a spherical panoramic image.
The Universal Transverse Mercator (UTM) is a map projection system for assigning coordinates to locations on the surface of the Earth. Like the traditional method of latitude and longitude, it is a horizontal position representation, which means it ignores altitude and treats the earth as a perfect ellipsoid. However, it differs from global latitude/longitude in that it divides earth into 60 zones and projects each to the plane as a basis for its coordinates. Specifying a location means specifying the zone and the x, y coordinate in that plane. The projection from spheroid to a UTM zone is some parameterization of the transverse Mercator projection. The parameters vary by nation or region or mapping system.
A Lambert conformal conic projection (LCC) is a conic map projection used for aeronautical charts, portions of the State Plane Coordinate System, and many national and regional mapping systems. It is one of seven projections introduced by Johann Heinrich Lambert in his 1772 publication Anmerkungen und Zusätze zur Entwerfung der Land und Himmelscharten.
The Earthcentered, Earthfixed coordinate system is a geographic and Cartesian coordinate system. It represents positions as X, Y, and Z coordinates. The origin is defined as the center of mass of Earth, hence the term geocentric Cartesian coordinates.
Local tangent plane coordinates (LTP), also known as local ellipsoidal system, local geodetic coordinate system, or local vertical, local horizontal coordinates (LVLH), are a geographical coordinate system based on the tangent plane defined by the local vertical direction and the Earth's axis of rotation. It consists of three coordinates: one represents the position along the northern axis, one along the local eastern axis, and one represents the vertical position. Two righthanded variants exist: east, north, up (ENU) coordinates and north, east, down (NED) coordinates. They serve for representing state vectors that are commonly used in aviation and marine cybernetics.
An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations.
Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a reference ellipsoid. They include geodetic latitude (north/south) ϕ, longitude (east/west) λ, and ellipsoidal heighth. The triad is also known as Earth ellipsoidal coordinates.
The equidistant conic projection is a conic map projection commonly used for maps of small countries as well as for larger regions such as the continental United States that are elongated easttowest.
Web Mercator, Google Web Mercator, Spherical Mercator, WGS 84 Web Mercator or WGS 84/PseudoMercator is a variant of the Mercator map projection and is the de facto standard for Web mapping applications. It rose to prominence when Google Maps adopted it in 2005. It is used by virtually all major online map providers, including Google Maps, Mapbox, Bing Maps, OpenStreetMap, Mapquest, Esri, and many others. Its official EPSG identifier is EPSG:3857, although others have been used historically.
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