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55-748: Coordinates : 43°01′17″N 76°41′20″W / 43.02139°N 76.68889°W / 43.02139; -76.68889 River in New York, United States Crane Brook [REDACTED] Location Country United States State New York Physical characteristics Mouth Seneca River • location Montezuma, New York , United States • coordinates 43°01′17″N 76°41′20″W / 43.02139°N 76.68889°W / 43.02139; -76.68889 Basin size 45.4 sq mi (118 km) Crane Brook flows into
110-502: A tan ϕ {\displaystyle \textstyle {\tan \beta ={\frac {b}{a}}\tan \phi }\,\!} ; for the GRS 80 and WGS 84 spheroids, b a = 0.99664719 {\textstyle {\tfrac {b}{a}}=0.99664719} . ( β {\displaystyle \textstyle {\beta }\,\!} is known as the reduced (or parametric) latitude ). Aside from rounding, this
165-456: A datum transformation such as a Helmert transformation , although in certain situations a simple translation may be sufficient. Datums may be global, meaning that they represent the whole Earth, or they may be local, meaning that they represent an ellipsoid best-fit to only a portion of the Earth. Examples of global datums include World Geodetic System (WGS 84, also known as EPSG:4326 ),
220-459: 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
275-423: 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
330-498: 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
385-588: A point on Earth's surface is the angle east or west of a reference meridian to another meridian that passes through that point. All meridians are halves of great ellipses (often called great circles ), which converge at the North and South Poles. The meridian of the British Royal Observatory in Greenwich , in southeast London, England, is the international prime meridian , although some organizations—such as
440-473: A region of the surface of the Earth. Some newer datums are bound to the center of mass of the Earth. This combination of mathematical model and physical binding mean that anyone using the same datum will obtain the same location measurement for the same physical location. However, two different datums will usually yield different location measurements for the same physical location, which may appear to differ by as much as several hundred meters; this not because
495-406: 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
550-428: 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
605-411: 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 ϕ {\displaystyle \phi } is where Earth's equatorial radius a {\displaystyle a} equals 6,378,137 m and tan β = b
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#1732858806164660-475: Is 110.6 km. The circles of longitude, meridians, meet at the geographical poles, with the west–east width of a second naturally decreasing as latitude increases. On the Equator at sea level, one longitudinal second measures 30.92 m, a longitudinal minute is 1855 m and a longitudinal degree is 111.3 km. At 30° a longitudinal second is 26.76 m, at Greenwich (51°28′38″N) 19.22 m, and at 60° it
715-519: Is 15.42 m. On the WGS 84 spheroid, the length in meters of a degree of latitude at latitude ϕ (that is, the number of meters you would have to travel along a north–south line to move 1 degree in latitude, when at latitude ϕ ), is about The returned measure of meters per degree latitude varies continuously with latitude. Similarly, the length in meters of a degree of longitude can be calculated as (Those coefficients can be improved, but as they stand
770-505: 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
825-449: 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
880-498: Is different from Wikidata Infobox mapframe without OSM relation ID on Wikidata Coordinates on Wikidata Pages using infobox river with mapframe Pages using the Kartographer extension Geographic coordinate system A geographic coordinate system ( GCS ) is a spherical or geodetic coordinate system for measuring and communicating positions directly on Earth as latitude and longitude . It
935-452: Is known as a graticule . The origin/zero point of this system is located in the Gulf of Guinea about 625 km (390 mi) south of Tema , Ghana , a location often facetiously called Null Island . In order to use the theoretical definitions of latitude, longitude, and height to precisely measure actual locations on the physical earth, a geodetic datum must be used. A horizonal datum
990-502: 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
1045-654: 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 m of each other if the two points are one degree of longitude apart. Like any series of multiple-digit numbers, latitude-longitude pairs can be challenging to communicate and remember. Therefore, alternative schemes have been developed for encoding GCS coordinates into alphanumeric strings or words: These are not distinct coordinate systems, only alternative methods for expressing latitude and longitude measurements. Earth ellipsoid An Earth ellipsoid or Earth spheroid
1100-409: Is the simplest, oldest and most widely used of the various spatial reference systems that are in use, and forms the basis for most others. Although latitude and longitude form a coordinate tuple like a cartesian coordinate system , the geographic coordinate system is not cartesian because the measurements are angles and are not on a planar surface. A full GCS specification, such as those listed in
1155-465: Is used to precisely measure latitude and longitude, while a vertical datum is used to measure elevation or altitude. Both types of datum bind a mathematical model of the shape of the earth (usually a reference ellipsoid for a horizontal datum, and a more precise geoid for a vertical datum) to the earth. Traditionally, this binding was created by a network of control points , surveyed locations at which monuments are installed, and were only accurate for
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#17328588061641210-741: The EPSG and ISO 19111 standards, also includes a choice of geodetic datum (including an Earth ellipsoid ), as different datums will yield different latitude and longitude values for the same location. The invention of a geographic coordinate system is generally credited to Eratosthenes of Cyrene , who composed his now-lost Geography at the Library of Alexandria in the 3rd century BC. A century later, Hipparchus of Nicaea improved on this system by determining latitude from stellar measurements rather than solar altitude and determining longitude by timings of lunar eclipses , rather than dead reckoning . In
1265-471: The International Date Line , which diverges from it in several places for political and convenience reasons, including between far eastern Russia and the far western Aleutian Islands . The combination of these two components specifies the position of any location on the surface of Earth, without consideration of altitude or depth. The visual grid on a map formed by lines of latitude and longitude
1320-538: The Seneca River by Montezuma, New York . References [ edit ] ^ "Crane brook" . usgs.gov . usgs. 1998 . Retrieved 30 May 2017 . data Retrieved from " https://en.wikipedia.org/w/index.php?title=Crane_Brook&oldid=1257507637 " Categories : Rivers of Cayuga County, New York Rivers of New York (state) Hidden categories: Pages using gadget WikiMiniAtlas Articles with short description Short description
1375-410: 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
1430-405: 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
1485-623: 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
1540-515: The 1st or 2nd century, Marinus of Tyre compiled an extensive gazetteer and mathematically plotted world map using coordinates measured east from a prime meridian at the westernmost known land, designated the Fortunate Isles , off the coast of western Africa around the Canary or Cape Verde Islands , and measured north or south of the island of Rhodes off Asia Minor . Ptolemy credited him with
1595-641: 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
1650-440: 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
1705-499: The Earth's surface move relative to each other due to continental plate motion, subsidence, and diurnal Earth tidal movement caused by the Moon and the Sun. This daily movement can be as much as a meter. Continental movement can be up to 10 cm a year, or 10 m in a century. A weather system high-pressure area can cause a sinking of 5 mm . Scandinavia is rising by 1 cm a year as a result of
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1760-708: The European ED50 , and the British OSGB36 . Given a location, the datum provides the latitude ϕ {\displaystyle \phi } and longitude λ {\displaystyle \lambda } . In the United Kingdom there are three common latitude, longitude, and height systems in use. WGS 84 differs at Greenwich from the one used on published maps OSGB36 by approximately 112 m. The military system ED50 , used by NATO , differs from about 120 m to 180 m. Points on
1815-524: The French Institut national de l'information géographique et forestière —continue to use other meridians for internal purposes. The prime meridian determines the proper Eastern and Western Hemispheres , although maps often divide these hemispheres further west in order to keep the Old World on a single side. The antipodal meridian of Greenwich is both 180°W and 180°E. This is not to be conflated with
1870-474: 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
1925-844: 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
1980-409: 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
2035-566: The center of the Earth. Lines joining points of the same latitude trace circles on the surface of Earth called parallels , as they are parallel to the Equator and to each other. The North Pole is 90° N; the South Pole is 90° S. The 0° parallel of latitude is designated the Equator , the fundamental plane of all geographic coordinate systems. The Equator divides the globe into Northern and Southern Hemispheres . The longitude λ of
2090-519: 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;
2145-613: The default datum used for the Global Positioning System , and the International Terrestrial Reference System and Frame (ITRF), used for estimating continental drift and crustal deformation . The distance to Earth's center can be used both for very deep positions and for positions in space. Local datums chosen by a national cartographical organization include the North American Datum ,
2200-454: 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
2255-490: 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 ϕ {\displaystyle \phi } 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 M r {\displaystyle \textstyle {M_{r}}\,\!}
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2310-637: 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
2365-413: 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
2420-468: The full adoption of longitude and latitude, rather than measuring latitude in terms of the length of the midsummer day. Ptolemy's 2nd-century Geography used the same prime meridian but measured latitude from the Equator instead. After their work was translated into Arabic in the 9th century, Al-Khwārizmī 's Book of the Description of the Earth corrected Marinus' and Ptolemy's errors regarding
2475-669: 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
2530-490: 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
2585-749: The length of the Mediterranean Sea , causing medieval Arabic cartography to use a prime meridian around 10° east of Ptolemy's line. Mathematical cartography resumed in Europe following Maximus Planudes ' recovery of Ptolemy's text a little before 1300; the text was translated into Latin at Florence by Jacopo d'Angelo around 1407. In 1884, the United States hosted the International Meridian Conference , attended by representatives from twenty-five nations. Twenty-two of them agreed to adopt
2640-461: The location has moved, but because the reference system used to measure it has shifted. Because any spatial reference system or map projection is ultimately calculated from latitude and longitude, it is crucial that they clearly state the datum on which they are based. For example, a UTM coordinate based on WGS84 will be different than a UTM coordinate based on NAD27 for the same location. Converting coordinates from one datum to another requires
2695-579: The longitude of the Royal Observatory in Greenwich , England as the zero-reference line. The Dominican Republic voted against the motion, while France and Brazil abstained. France adopted Greenwich Mean Time in place of local determinations by the Paris Observatory in 1911. The latitude ϕ of a point on Earth's surface is the angle between the equatorial plane and the straight line that passes through that point and through (or close to)
2750-460: The melting of the ice sheets of the last ice age , but neighboring Scotland is rising by only 0.2 cm . These changes are insignificant if a local datum is used, but are statistically significant if a global datum is used. On the GRS 80 or WGS 84 spheroid at sea level at the Equator, one latitudinal second measures 30.715 m , one latitudinal minute is 1843 m and one latitudinal degree
2805-493: 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
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#17328588061642860-435: 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
2915-472: 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
2970-568: 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
3025-399: 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
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