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70-458: GEOREF is based on the standard system of latitude and longitude , but uses a simpler and more concise notation. GEOREF divides the Earth's surface into successively smaller quadrangles, with a notation system used to identify each quadrangle within its parent. Unlike latitude/longitude, GEOREF runs in one direction horizontally, east from the 180° meridian ; and one direction vertically, north from

140-484: A geographic coordinate system as defined in the specification of the ISO 19111 standard. Since there are many different reference ellipsoids , the precise latitude of a feature on the surface is not unique: this is stressed in the ISO standard which states that "without the full specification of the coordinate reference system, coordinates (that is latitude and longitude) are ambiguous at best and meaningless at worst". This

210-459: A system of linear equations formulated via linearization of M {\displaystyle M} : where the partial derivatives are: Longer arcs with multiple intermediate-latitude determinations can completely determine the ellipsoid that best fits the surveyed region. In practice, multiple arc measurements are used to determine the ellipsoid parameters by the method of least squares adjustment . The parameters determined are usually

280-560: A 300-by-300-pixel sphere, so illustrations usually exaggerate the flattening. The graticule on the ellipsoid is constructed in exactly the same way as on the sphere. The normal at a point on the surface of an ellipsoid does not pass through the centre, except for points on the equator or at the poles, but the definition of latitude remains unchanged as the angle between the normal and the equatorial plane. The terminology for latitude must be made more precise by distinguishing: Geographic latitude must be used with care, as some authors use it as

350-423: A difference of the major and minor semi-axes of approximately 21 km (13 miles) (more precisely, 21.3846857548205 km). For comparison, Earth's Moon is even less elliptical, with a flattening of less than 1/825, while Jupiter is visibly oblate at about 1/15 and one of Saturn's triaxial moons, Telesto , is highly flattened, with f between 1/3 and 1/2 (meaning that the polar diameter is between 50% and 67% of

420-475: A height of 17,000 feet over GJPJ3716. Latitude In geography , latitude is a coordinate that specifies the north – south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from −90° at the south pole to 90° at the north pole, with 0° at the Equator . Lines of constant latitude , or parallels , run east–west as circles parallel to

490-507: A mathematical reference surface, this surface should have a similar curvature as the regional geoid; otherwise, reduction of the measurements will get small distortions. This is the reason for the "long life" of former reference ellipsoids like the Hayford or the Bessel ellipsoid , despite the fact that their main axes deviate by several hundred meters from the modern values. Another reason

560-410: A restricted geographical area). For example, on a GEOREF chart, Naval Air Station Patuxent River (38°17′10″N 76°24′42″W) / (38.286108, -76.4291704) is located (to the nearest minute) at position GJPJ3417. [REDACTED] To locate the position from the coordinates, proceed as follows: The same co-ordinate shown in 6-digit (1/10 minute) format is GJPJ342171 and in 8-digit (1/100 minute) format

630-406: A shape which he termed an oblate spheroid . In geophysics, geodesy , and related areas, the word 'ellipsoid' is understood to mean 'oblate ellipsoid of revolution', and the older term 'oblate spheroid' is hardly used. For bodies that cannot be well approximated by an ellipsoid of revolution a triaxial (or scalene) ellipsoid is used. The shape of an ellipsoid of revolution is determined by

700-443: A survey but, with the advent of GPS , it has become natural to use reference ellipsoids (such as WGS84 ) with centre at the centre of mass of the Earth and minor axis aligned to the rotation axis of the Earth. These geocentric ellipsoids are usually within 100 m (330 ft) of the geoid. Since latitude is defined with respect to an ellipsoid, the position of a given point is different on each ellipsoid: one cannot exactly specify

770-555: A synonym for geodetic latitude whilst others use it as an alternative to the astronomical latitude . "Latitude" (unqualified) should normally refer to the geodetic latitude. The importance of specifying the reference datum may be illustrated by a simple example. On the reference ellipsoid for WGS84, the centre of the Eiffel Tower has a geodetic latitude of 48° 51′ 29″ N, or 48.8583° N and longitude of 2° 17′ 40″ E or 2.2944°E. The same coordinates on

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840-428: A theoretical coherence between the geographic latitude and the meridional curvature of the geoid . The latter is close to the mean sea level , and therefore an ideal Earth ellipsoid has the same volume as the geoid. While the mean Earth ellipsoid is the ideal basis of global geodesy, for regional networks a so-called reference ellipsoid may be the better choice. When geodetic measurements have to be computed on

910-479: Is GJPJ34241716. Extensions to the above notation allow the GEOREF system to be used to designate an area around a reference point. This is achieved by adding an area designation to a base GEOREF co-ordinate. The area designation can be the letter S, to specify the sides of a rectangle (separated by the letter X); or the letter R, to specify the radius of a circle. In both cases the units are nautical miles. In addition,

980-520: Is a judicial one: the coordinates of millions of boundary stones should remain fixed for a long period. If their reference surface changes, the coordinates themselves also change. However, for international networks, GPS positioning, or astronautics , these regional reasons are less relevant. As knowledge of the Earth's figure is increasingly accurate, the International Geoscientific Union IUGG usually adapts

1050-453: 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. It is a spheroid (an ellipsoid of revolution ) whose minor axis (shorter diameter), which connects the geographical North Pole and South Pole , is approximately aligned with

1120-411: Is also used in the current literature. The parametric latitude is related to the geodetic latitude by: The alternative name arises from the parameterization of the equation of the ellipse describing a meridian section. In terms of Cartesian coordinates p , the distance from the minor axis, and z , the distance above the equatorial plane, the equation of the ellipse is: The Cartesian coordinates of

1190-484: Is determined by the shape of the ellipse which is rotated about its minor (shorter) axis. Two parameters are required. One is invariably the equatorial radius, which is the semi-major axis , a . The other parameter is usually (1) the polar radius or semi-minor axis , b ; or (2) the (first) flattening , f ; or (3) the eccentricity , e . These parameters are not independent: they are related by Many other parameters (see ellipse , ellipsoid ) appear in

1260-453: Is determined with the meridian altitude method. More precise measurement of latitude requires an understanding of the gravitational field of the Earth, either to set up theodolites or to determine GPS satellite orbits. The study of the figure of the Earth together with its gravitational field is the science of geodesy . The graticule is formed by the lines of constant latitude and constant longitude, which are constructed with reference to

1330-466: Is of great importance in accurate applications, such as a Global Positioning System (GPS), but in common usage, where high accuracy is not required, the reference ellipsoid is not usually stated. In English texts, the latitude angle, defined below, is usually denoted by the Greek lower-case letter phi ( ϕ or φ ). It is measured in degrees , minutes and seconds or decimal degrees , north or south of

1400-507: Is part of a more encompassing geodetic datum . For example, the older ED-50 ( European Datum 1950 ) is based on the Hayford or International Ellipsoid . WGS-84 is peculiar in that the same name is used for both the complete geodetic reference system and its component ellipsoidal model. Nevertheless, the two concepts—ellipsoidal model and geodetic reference system—remain distinct. Note that the same ellipsoid may be known by different names. It

1470-451: Is the angle between the equatorial plane and the normal to the surface at that point: the normal to the surface of the sphere is along the radial vector. The latitude, as defined in this way for the sphere, is often termed the spherical latitude, to avoid ambiguity with the geodetic latitude and the auxiliary latitudes defined in subsequent sections of this article. Besides the equator, four other parallels are of significance: The plane of

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1540-421: Is the meridional radius of curvature . The quarter meridian distance from the equator to the pole is For WGS84 this distance is 10 001 .965 729  km . The evaluation of the meridian distance integral is central to many studies in geodesy and map projection. It can be evaluated by expanding the integral by the binomial series and integrating term by term: see Meridian arc for details. The length of

1610-522: The Philosophiæ Naturalis Principia Mathematica , in which he proved that a rotating self-gravitating fluid body in equilibrium takes the form of an oblate ellipsoid. (This article uses the term ellipsoid in preference to the older term spheroid .) Newton's result was confirmed by geodetic measurements in the 18th century. (See Meridian arc .) An oblate ellipsoid is the three-dimensional surface generated by

1680-485: The South Pole . GEOREF can easily be adapted to give co-ordinates with varying degrees of precision, using a 2–12 character geocode . GEOREF co-ordinates are defined by successive divisions of the Earth's surface, as follows: The initial two letters of a GEOREF reference, designating the 15 degree quadrangle, can be omitted, if it is clear which 15 degree quadrangle the reference applies to (e.g., when working within

1750-410: The flattening f , defined as: That is, f is the amount of flattening at each pole, relative to the radius at the equator. This is often expressed as a fraction 1/ m ; m = 1/ f then being the "inverse flattening". A great many other ellipse parameters are used in geodesy but they can all be related to one or two of the set a , b and f . A great many ellipsoids have been used to model

1820-405: The interior , as well as the subsequent flattening caused by the centrifugal force from the rotation of these massive objects (for planetary bodies that do rotate). 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. In

1890-580: The zenith ). On map projections there is no universal rule as to how meridians and parallels should appear. The examples below show the named parallels (as red lines) on the commonly used Mercator projection and the Transverse Mercator projection . On the former the parallels are horizontal and the meridians are vertical, whereas on the latter there is no exact relationship of parallels and meridians with horizontal and vertical: both are complicated curves. \ In 1687 Isaac Newton published

1960-646: The 1967 meeting of the IUGG held in Lucerne, Switzerland, the ellipsoid called GRS-67 ( Geodetic Reference System 1967) in the listing was recommended for adoption. The new ellipsoid was not recommended to replace the International Ellipsoid (1924), but was advocated for use where a greater degree of accuracy is required. It became a part of the GRS-67 which was approved and adopted at the 1971 meeting of

2030-641: The Earth in the past, with different assumed values of a and b as well as different assumed positions of the center and different axis orientations relative to the solid Earth. Starting in the late twentieth century, improved measurements of satellite orbits and star positions have provided extremely accurate determinations of the Earth's center of mass and of its axis of revolution; and those parameters have been adopted also for all modern reference ellipsoids. The ellipsoid WGS-84 , widely used for mapping and satellite navigation has f close to 1/300 (more precisely, 1/298.257223563, by definition), corresponding to

2100-440: The Earth's axis of rotation. The ellipsoid is defined by the equatorial axis ( a ) and the polar axis ( b ); their radial difference is slightly more than 21 km, or 0.335% of a (which is not quite 6,400 km). Many methods exist for determination of the axes of an Earth ellipsoid, ranging from meridian arcs up to modern satellite geodesy or the analysis and interconnection of continental geodetic networks . Amongst

2170-450: The Earth's orbit about the Sun is called the ecliptic , and the plane perpendicular to the rotation axis of the Earth is the equatorial plane. The angle between the ecliptic and the equatorial plane is called variously the axial tilt, the obliquity, or the inclination of the ecliptic, and it is conventionally denoted by i . The latitude of the tropical circles is equal to i and the latitude of

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2240-478: The GRS-80 flattening because the normalized second degree zonal harmonic gravitational coefficient, that was derived from the GRS-80 value for J 2 {\displaystyle J_{2}} , was truncated to eight significant digits in the normalization process. An ellipsoidal model describes only the ellipsoid's geometry and a normal gravity field formula to go with it. Commonly an ellipsoidal model

2310-873: The IUGG held in Moscow. It is used in Australia for the Australian Geodetic Datum and in the South American Datum 1969. The GRS-80 (Geodetic Reference System 1980) as approved and adopted by the IUGG at its Canberra, Australia meeting of 1979 is based on the equatorial radius (semi-major axis of Earth ellipsoid) a {\displaystyle a} , total mass G M {\displaystyle GM} , dynamic form factor J 2 {\displaystyle J_{2}} and angular velocity of rotation ω {\displaystyle \omega } , making

2380-691: The Sun is overhead at some point of the Tropic of Capricorn . The south polar latitudes below the Antarctic Circle are in daylight, whilst the north polar latitudes above the Arctic Circle are in night. The situation is reversed at the June solstice, when the Sun is overhead at the Tropic of Cancer. Only at latitudes in between the two tropics is it possible for the Sun to be directly overhead (at

2450-571: The WGS84 spheroid is The variation of this distance with latitude (on WGS84 ) is shown in the table along with the length of a degree of longitude (east–west distance): A calculator for any latitude is provided by the U.S. Government's National Geospatial-Intelligence Agency (NGA). The following graph illustrates the variation of both a degree of latitude and a degree of longitude with latitude. There are six auxiliary latitudes that have applications to special problems in geodesy, geophysics and

2520-425: The angle subtended at the centre by the meridian arc from the equator to the point concerned. If the meridian distance is denoted by m ( ϕ ) then where R denotes the mean radius of the Earth. R is equal to 6,371 km or 3,959 miles. No higher accuracy is appropriate for R since higher-precision results necessitate an ellipsoid model. With this value for R the meridian length of 1 degree of latitude on

2590-409: The axes of the Earth ellipsoid to the best available data. 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

2660-526: The centre of the Earth and perpendicular to the rotation axis intersects the surface at a great circle called the Equator . Planes parallel to the equatorial plane intersect the surface in circles of constant latitude; these are the parallels. The Equator has a latitude of 0°, the North Pole has a latitude of 90° North (written 90° N or +90°), and the South Pole has a latitude of 90° South (written 90° S or −90°). The latitude of an arbitrary point

2730-524: The context of standardization and geographic applications, a geodesic reference ellipsoid is the mathematical model used as foundation by spatial reference system or geodetic datum definitions. In 1687 Isaac Newton published the Principia in which he included a proof that a rotating self-gravitating fluid body in equilibrium takes the form of a flattened ("oblate") ellipsoid of revolution, generated by an ellipse rotated around its minor diameter;

2800-408: The datum ED50 define a point on the ground which is 140 metres (460 feet) distant from the tower. A web search may produce several different values for the latitude of the tower; the reference ellipsoid is rarely specified. The length of a degree of latitude depends on the figure of the Earth assumed. On the sphere the normal passes through the centre and the latitude ( ϕ ) is therefore equal to

2870-529: The definitions of latitude and longitude. In the first step the physical surface is modeled by the geoid , a surface which approximates the mean sea level over the oceans and its continuation under the land masses. The second step is to approximate the geoid by a mathematically simpler reference surface. The simplest choice for the reference surface is a sphere , but the geoid is more accurately modeled by an ellipsoid of revolution . The definitions of latitude and longitude on such reference surfaces are detailed in

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2940-458: The different set of data used in national surveys are several of special importance: the Bessel ellipsoid of 1841, the international Hayford ellipsoid of 1924, and (for GPS positioning) the WGS84 ellipsoid. There are two types of ellipsoid: mean and reference. A data set which describes the global average of the Earth's surface curvature is called the mean Earth Ellipsoid . It refers to

3010-402: The ellipsoid to that point Q on the surrounding sphere (of radius a ) which is the projection parallel to the Earth's axis of a point P on the ellipsoid at latitude ϕ . It was introduced by Legendre and Bessel who solved problems for geodesics on the ellipsoid by transforming them to an equivalent problem for spherical geodesics by using this smaller latitude. Bessel's notation, u ( ϕ ) ,

3080-517: The equator. For navigational purposes positions are given in degrees and decimal minutes. For instance, The Needles lighthouse is at 50°39.734′ N 001°35.500′ W. This article relates to coordinate systems for the Earth: it may be adapted to cover the Moon, planets and other celestial objects ( planetographic latitude ). For a brief history, see History of latitude . In celestial navigation , latitude

3150-452: The equator. Latitude and longitude are used together as a coordinate pair to specify a location on the surface of the Earth. On its own, the term "latitude" normally refers to the geodetic latitude as defined below. Briefly, the geodetic latitude of a point is the angle formed between the vector perpendicular (or normal ) to the ellipsoidal surface from the point, and the plane of the equator . Two levels of abstraction are employed in

3220-637: The equatorial radius a 0 {\displaystyle a_{0}} and for the flattening f 0 {\displaystyle f_{0}} . The theoretical Earth's meridional radius of curvature M 0 ( φ i ) {\displaystyle M_{0}(\varphi _{i})} can be calculated at the latitude of each arc measurement as: where e 0 2 = 2 f 0 − f 0 2 {\displaystyle e_{0}^{2}=2f_{0}-f_{0}^{2}} . Then discrepancies between empirical and theoretical values of

3290-413: The equatorial. Arc measurement is the historical method of determining the ellipsoid. Two meridian arc measurements will allow the derivation of two parameters required to specify a reference ellipsoid. For example, if the measurements were hypothetically performed exactly over the equator plane and either geographical pole, the radii of curvature so obtained would be related to the equatorial radius and

3360-438: The following sections. Lines of constant latitude and longitude together constitute a graticule on the reference surface. The latitude of a point on the actual surface is that of the corresponding point on the reference surface, the correspondence being along the normal to the reference surface, which passes through the point on the physical surface. Latitude and longitude together with some specification of height constitute

3430-399: The geocentric latitude ( θ ) and the geodetic latitude ( ϕ ) is: For points not on the surface of the ellipsoid, the relationship involves additionally the ellipsoidal height h : where N is the prime vertical radius of curvature. The geodetic and geocentric latitudes are equal at the equator and at the poles but at other latitudes they differ by a few minutes of arc. Taking the value of

3500-696: The individual who derived them and the year of development is given. In 1887 the English surveyor Colonel Alexander Ross Clarke CB FRS RE was awarded the Gold Medal of the Royal Society for his work in determining the figure of the Earth. The international ellipsoid was developed by John Fillmore Hayford in 1910 and adopted by the International Union of Geodesy and Geophysics (IUGG) in 1924, which recommended it for international use. At

3570-494: The inverse flattening 1 / f {\displaystyle 1/f} a derived quantity. The minute difference in 1 / f {\displaystyle 1/f} seen between GRS-80 and WGS-84 results from an unintentional truncation in the latter's defining constants: while the WGS-84 was designed to adhere closely to the GRS-80, incidentally the WGS-84 derived flattening turned out to differ slightly from

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3640-451: The latitude and longitude of a geographical feature without specifying the ellipsoid used. Many maps maintained by national agencies are based on older ellipsoids, so one must know how the latitude and longitude values are transformed from one ellipsoid to another. GPS handsets include software to carry out datum transformations which link WGS84 to the local reference ellipsoid with its associated grid. The shape of an ellipsoid of revolution

3710-415: The letter H can be added, followed by an altitude in thousands of feet. For example, the reference GJQJ0207S6X8 designates a rectangle centered on Deal Island (GJQJ0207), running 6 nautical miles (11 km) east–west and 8 nautical miles (15 km) north–south. Designation GJPJ4103R5 means a circle around Point Lookout (GJPJ4103) with a radius of 5 nautical miles (9 km). Designation GJPJ3716H17 means

3780-538: The meridian arc between two given latitudes is given by replacing the limits of the integral by the latitudes concerned. The length of a small meridian arc is given by When the latitude difference is 1 degree, corresponding to ⁠ π / 180 ⁠ radians, the arc distance is about The distance in metres (correct to 0.01 metre) between latitudes ϕ {\displaystyle \phi }  − 0.5 degrees and ϕ {\displaystyle \phi }  + 0.5 degrees on

3850-493: The methods of satellite geodesy , especially satellite gravimetry . 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 height h (also known as geodetic height ). The reference ellipsoid models listed below have had utility in geodetic work and many are still in use. The older ellipsoids are named for

3920-521: The point are parameterized by Cayley suggested the term parametric latitude because of the form of these equations. The parametric latitude is not used in the theory of map projections. Its most important application is in the theory of ellipsoid geodesics, ( Vincenty , Karney ). The rectifying latitude , μ , is the meridian distance scaled so that its value at the poles is equal to 90 degrees or ⁠ π / 2 ⁠ radians: Reference ellipsoid An Earth ellipsoid or Earth spheroid

3990-463: The polar circles is its complement (90° - i ). The axis of rotation varies slowly over time and the values given here are those for the current epoch . The time variation is discussed more fully in the article on axial tilt . The figure shows the geometry of a cross-section of the plane perpendicular to the ecliptic and through the centres of the Earth and the Sun at the December solstice when

4060-435: The polar radius, respectively a and b (see: Earth polar and equatorial radius of curvature ). Then, the flattening would readily follow from its definition: For two arc measurements each at arbitrary average latitudes φ i {\displaystyle \varphi _{i}} , i = 1 , 2 {\displaystyle i=1,\,2} , the solution starts from an initial approximation for

4130-472: The radius of curvature can be formed as δ M i = M i − M 0 ( φ i ) {\displaystyle \delta M_{i}=M_{i}-M_{0}(\varphi _{i})} . Finally, corrections for the initial equatorial radius δ a {\displaystyle \delta a} and the flattening δ f {\displaystyle \delta f} can be solved by means of

4200-509: The reference ellipsoid to the plane or in calculations of geodesics on the ellipsoid. Their numerical values are not of interest. For example, no one would need to calculate the authalic latitude of the Eiffel Tower. The expressions below give the auxiliary latitudes in terms of the geodetic latitude, the semi-major axis, a , and the eccentricity, e . (For inverses see below .) The forms given are, apart from notational variants, those in

4270-473: The rotation axis of the Earth. The primary reference points are the poles where the axis of rotation of the Earth intersects the reference surface. Planes which contain the rotation axis intersect the surface at the meridians ; and the angle between any one meridian plane and that through Greenwich (the Prime Meridian ) defines the longitude: meridians are lines of constant longitude. The plane through

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4340-417: The rotation of an ellipse about its shorter axis (minor axis). "Oblate ellipsoid of revolution" is abbreviated to 'ellipsoid' in the remainder of this article. (Ellipsoids which do not have an axis of symmetry are termed triaxial .) Many different reference ellipsoids have been used in the history of geodesy . In pre-satellite days they were devised to give a good fit to the geoid over the limited area of

4410-468: The semi-major axis and the inverse flattening, ⁠ 1 / f ⁠ . For example, the defining values for the WGS84 ellipsoid, used by all GPS devices, are from which are derived The difference between the semi-major and semi-minor axes is about 21 km (13 miles) and as fraction of the semi-major axis it equals the flattening; on a computer monitor the ellipsoid could be sized as 300 by 299 pixels. This would barely be distinguishable from

4480-568: The semi-major axis, a {\displaystyle a} , and any of the semi-minor axis, b {\displaystyle b} , flattening , or eccentricity. Regional-scale systematic effects observed in the radius of curvature measurements reflect the geoid undulation and the deflection of the vertical , as explored in astrogeodetic leveling . Gravimetry is another technique for determining Earth's flattening, as per Clairaut's theorem . Modern geodesy no longer uses simple meridian arcs or ground triangulation networks, but

4550-399: The shape parameters of that ellipse . The semi-major axis of the ellipse, a , becomes the equatorial radius of the ellipsoid: the semi-minor axis of the ellipse, b , becomes the distance from the centre to either pole. These two lengths completely specify the shape of the ellipsoid. In geodesy publications, however, it is common to specify the semi-major axis (equatorial radius) a and

4620-420: The sphere is 111.2 km (69.1 statute miles) (60.0 nautical miles). The length of one minute of latitude is 1.853 km (1.151 statute miles) (1.00 nautical miles), while the length of 1 second of latitude is 30.8 m or 101 feet (see nautical mile ). In Meridian arc and standard texts it is shown that the distance along a meridian from latitude ϕ to the equator is given by ( ϕ in radians) where M ( ϕ )

4690-403: The squared eccentricity as 0.0067 (it depends on the choice of ellipsoid) the maximum difference of ϕ − θ {\displaystyle \phi {-}\theta } may be shown to be about 11.5 minutes of arc at a geodetic latitude of approximately 45° 6′. The parametric latitude or reduced latitude , β , is defined by the radius drawn from the centre of

4760-458: The standard reference for map projections, namely "Map projections: a working manual" by J. P. Snyder. Derivations of these expressions may be found in Adams and online publications by Osborne and Rapp. The geocentric latitude is the angle between the equatorial plane and the radius from the centre to a point of interest. When the point is on the surface of the ellipsoid, the relation between

4830-472: The study of geodesy, geophysics and map projections but they can all be expressed in terms of one or two members of the set a , b , f and e . Both f and e are small and often appear in series expansions in calculations; they are of the order ⁠ 1 / 298 ⁠ and 0.0818 respectively. Values for a number of ellipsoids are given in Figure of the Earth . Reference ellipsoids are usually defined by

4900-407: The theory of map projections: The definitions given in this section all relate to locations on the reference ellipsoid but the first two auxiliary latitudes, like the geodetic latitude, can be extended to define a three-dimensional geographic coordinate system as discussed below . The remaining latitudes are not used in this way; they are used only as intermediate constructs in map projections of

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