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The kind of [[figure (geometry)|figure]] depends on application, including the precision needed for the model. A [[spherical Earth]] is a well-known historical approximation that is satisfactory for [[geography]], [[astronomy]] and many other purposes. Several models with greater accuracy (including [[Earth ellipsoid|ellipsoid]]) have been developed so that [[Geographic coordinate system|coordinate systems]] can serve the precise needs of [[navigation]], [[surveying]], [[cadastre]], [[land use]], and various other concerns. == Motivation == Earth's [[Topography|topographic]] surface is apparent with its variety of land forms and water areas. This topographic surface is generally the concern of topographers, [[Hydrography|hydrographers]], and [[Geophysics|geophysicists]]. While it is the surface on which Earth measurements are made, mathematically modeling it while taking the irregularities into account would be extremely complicated. The [[Pythagoreanism|Pythagorean]] concept of a [[spherical Earth]] offers a simple surface that is easy to deal with mathematically. Many astronomical and navigational computations use a [[sphere]] to model the Earth as a close approximation. However, a more accurate figure is needed for measuring distances and areas on the scale beyond the purely local. Better approximations can be made by modeling the entire surface as an [[oblate spheroid]], using [[spherical harmonic]]s to approximate the [[geoid]], or modeling a region with a best-fit [[reference ellipsoid]]. For surveys of small areas, a planar (flat) model of Earth's surface suffices because the local topography overwhelms the curvature. [[Plane-table]] surveys are made for relatively small areas without considering the size and shape of the entire Earth. A survey of a city, for example, might be conducted this way. [[File:Earth2014shape SouthAmerica small.jpg|thumb|Topographic view of Earth relative to Earth's center (instead of to [[mean sea level]], as in common topographic maps)]] By the late 1600s, serious effort was devoted to modeling the Earth as an ellipsoid, beginning with French astronomer [[Jean Picard]]'s measurement of a degree of arc along the [[Paris meridian]]. Improved maps and better measurement of distances and areas of national territories motivated these early attempts. Surveying instrumentation and techniques improved over the ensuing centuries. Models for the figure of the Earth improved in step. In the mid- to late 20th century, research across the [[Earth science|geosciences]] contributed to drastic improvements in the accuracy of the figure of the Earth. The primary utility of this improved accuracy was to provide geographical and gravitational data for the [[inertial guidance system]]s of [[ballistic missile]]s. This funding also drove the expansion of geoscientific disciplines, fostering the creation and growth of various geoscience departments at many universities.<ref>{{cite journal |last=Cloud |first=John |title=Crossing the Olentangy River: The Figure of the Earth and the Military-Industrial-Academic Complex, 1947–1972 |journal=Studies in History and Philosophy of Modern Physics |volume=31 |issue=3 |pages=371–404 |year=2000 |doi=10.1016/S1355-2198(00)00017-4 |bibcode=2000SHPMP..31..371C }}</ref> These developments benefited many civilian pursuits as well, such as weather and communication [[satellite]] control and [[GPS]] location-finding, which would be impossible without highly accurate models for the figure of the Earth. == Models == The models for the figure of the Earth vary in the way they are used, in their complexity, and in the accuracy with which they represent the size and shape of the Earth. === Sphere === {{Main|Spherical Earth}} {{Further|Earth radius|Earth's circumference}} [[File:Horizon, Valencia (Spain).JPG|thumb|upright=1.2|A view across a 20-km-wide bay in the coast of [[Spain]]. The curvature of the Earth is evident in the [[horizon]] across the image, and the bases of the buildings on the far shore are below that horizon and hidden by the sea.]] The simplest model for the shape of the entire Earth is a sphere. The Earth's [[radius]] is the [[distance]] from Earth's center to its surface, about {{cvt|6371|km|mi|sp=us}}. While "radius" normally is a characteristic of perfect spheres, the Earth deviates from spherical by only a third of a percent, sufficiently close to treat it as a sphere in many contexts and justifying the term "the radius of the Earth". The concept of a spherical Earth dates back to around the [[6th century BC]],<ref>{{cite book |last=Dicks |first=D.R. |title=Early Greek Astronomy to Aristotle |pages=[https://archive.org/details/earlygreekastron0000dick/page/72 72–198] |year=1970 |isbn=978-0-8014-0561-7 |publisher=[[Cornell University Press]] |location=Ithaca, N.Y. |url=https://archive.org/details/earlygreekastron0000dick/page/72 }}</ref> but remained a matter of philosophical speculation until the [[3rd century BC]]. The first scientific estimation of the radius of the Earth was given by [[Eratosthenes]] about 240&nbsp;BC, with estimates of the accuracy of Eratosthenes's measurement ranging from −1% to 15%. The Earth is only approximately spherical, so no single value serves as its natural radius. Distances from points on the surface to the center range from {{cvt|6,353|km|mi}} to {{cvt|6,384|km|mi}}. Several different ways of modeling the Earth as a sphere each yield a mean radius of {{cvt|6371|km|mi|sp=us}}. Regardless of the model, any radius falls between the polar minimum of about {{cvt|6,357|km|mi}} and the equatorial maximum of about {{cvt|6,378|km|mi}}. The difference {{cvt|21|km|mi|sp=us}} correspond to the polar radius being approximately 0.3% shorter than the equatorial radius. ===Ellipsoid of revolution=== [[File:OblateSpheroid.PNG|thumb|An [[oblate spheroid]], highly exaggerated relative to the actual Earth]] [[File:Earth oblateness to scale.svg|thumb|A scale diagram of the [[flattening|oblateness]] of the 2003 [[IERS]] [[reference ellipsoid]], with north at the top. The outer edge of the dark blue line is an [[ellipse]] with the same [[Eccentricity (mathematics)#Ellipses|eccentricity]] as that of Earth. For comparison, the light blue circle within has a diameter equal to the ellipse's [[minor axis]]. The red curve represents the [[Karman line]] {{convert|100|km|mi|abbr=on}} above [[sea level]], while the yellow band denotes the [[apsis|altitude]] range of the [[International Space Station|ISS]] in [[low Earth orbit]].]] {{main|Earth ellipsoid}} {{Further|Reference ellipsoid}} As theorized by [[Isaac Newton]] and [[Christiaan Huygens]],<ref name="DMA" />{{Rp|page=4}} the Earth is [[flattening|flattened]] at the poles and [[equatorial bulge|bulged]] at the [[equator]]. Thus, [[geodesy]] represents the figure of the Earth as an oblate [[spheroid]]. The oblate spheroid, or [[oblate ellipsoid]], is an [[ellipsoid of revolution]] obtained by rotating an ellipse about its shorter axis. It is the regular [[geometric]] shape that most nearly approximates the shape of the Earth. A spheroid describing the figure of the Earth or other [[celestial body]] is called a [[reference ellipsoid]]. The reference ellipsoid for Earth is called an ''Earth ellipsoid''. An ellipsoid of revolution is uniquely defined by two quantities. Several conventions for expressing the two quantities are used in geodesy, but they are all equivalent to and convertible with each other: * Equatorial radius <math>a</math> (called ''semimajor axis''), and polar radius <math>b</math> (called ''semiminor axis''); * <math>a</math> and [[Eccentricity (mathematics)|eccentricity]] <math>e</math>; * <math>a</math> and flattening <math>f</math>. Eccentricity and flattening are different ways of expressing how squashed the ellipsoid is. When flattening appears as one of the defining quantities in geodesy, generally it is expressed by its reciprocal. For example, in the [[World Geodetic System#WGS84|WGS 84]] spheroid used by today's GPS systems, the reciprocal of the flattening <math>1/f</math> is set to be exactly {{gaps|298.257|223|563}}. The difference between a sphere and a reference ellipsoid for Earth is small, only about one part in 300. Historically, flattening was computed from [[grade measurement]]s. Nowadays, geodetic networks and [[satellite geodesy]] are used. In practice, many reference ellipsoids have been developed over the centuries from different surveys. The flattening value varies slightly from one reference ellipsoid to another, reflecting local conditions and whether the reference ellipsoid is intended to model the entire Earth or only some portion of it. A sphere has a single [[radius of curvature (applications)|radius of curvature]], which is simply the radius of the sphere. More complex surfaces have radii of curvature that vary over the surface. The radius of curvature describes the radius of the sphere that best approximates the surface at that point. Oblate ellipsoids have a constant radius of curvature east to west along [[Parallel (latitude)|parallels]], if a [[Geographic coordinate system|graticule]] is drawn on the surface, but varying curvature in any other direction. For an oblate ellipsoid, the polar radius of curvature <math>r_p</math> is larger than the equatorial :<math> r_p=\frac{a^2}{b},</math> because the pole is flattened: the flatter the surface, the larger the sphere must be to approximate it. Conversely, the ellipsoid's north–south radius of curvature at the equator <math>r_e</math> is smaller than the polar :<math> r_e=\frac{b^2}{a}</math> where <math>a</math> is the distance from the center of the ellipsoid to the equator (semi-major axis), and <math>b</math> is the distance from the center to the pole. (semi-minor axis) === Non-spheroidal deviations === ==== Triaxiality (equatorial eccentricity) {{anchor|Triaxiality|Equatorial eccentricity}} ==== The possibility that the Earth's equator is better characterized as an ellipse rather than a circle and therefore that the ellipsoid is triaxial has been a matter of scientific inquiry for many years.<ref>{{cite journal | title=Is the Earth a triaxial ellipsoid? | journal=Journal of Geophysical Research | date=1962 | volume=67 | issue=1 | pages=321–327 | first=W. A. | last=Heiskanen|bibcode = 1962JGR....67..321H |doi = 10.1029/JZ067i001p00321 }}</ref><ref>{{cite journal | title=Parameters of the Earth's tri-axial level ellipsoid | journal=Studia Geophysica et Geodaetica | date=1993 | volume=37 | issue=1 | pages=1–13 | first=Milan | last=Burša|bibcode = 1993StGG...37....1B |doi = 10.1007/BF01613918 | s2cid=128674427 }}</ref> Modern technological developments have furnished new and rapid methods for data collection and, since the launch of [[Sputnik 1]], orbital data have been used to investigate the theory of ellipticity.<ref name="DMA">{{cite report |url=https://apps.dtic.mil/sti/citations/ADA142764 |title=Geodesy for the Layman |author=Defense Mapping Agency |date=1983 |publisher=United States Air Force |edition=4th}}</ref> More recent results indicate a 70&nbsp;m difference between the two equatorial major and minor axes of inertia, with the larger semidiameter pointing to 15° W longitude (and also 180-degree away).<ref>Torge & Müller (2012) Geodesy, De Gruyter, p.100</ref><ref>Marchenko, A.N. (2009): Current estimation of the Earth’s mechanical and geometrical para meters. In Sideris, M.G., ed. (2009): Observing our changing Earth. IAG Symp. Proceed. 133., pp. 473–481. DOI:10.1007/978-3-540-85426-5_57</ref> {{Anchor|Pear shape}} ==== Egg or pear shape ==== Following work by Picard, Italian polymath [[Giovanni Domenico Cassini]] found that the length of a degree was apparently shorter north of Paris than to the south, implying the Earth to be [[egg]]-shaped.<ref name="DMA" />{{Rp|page=4}} In 1498, [[Christopher Columbus]] dubiously suggested that the Earth was pear-shaped based on his disparate mobile readings of the angle of the [[North Star]], which he incorrectly interpreted as having varying [[diurnal motion]].<ref>{{Cite book |last=Morison |first=Samuel Eliot |author-link=Samuel Eliot Morison |title=Admiral of the Ocean Sea: A Life of Christopher Columbus |publisher=Little, Brown and Company |year=1991 |isbn=978-0-316-58478-4 |location=Boston |page=557 |oclc=1154365097 |orig-year=1942}}</ref> The theory of a slightly pear-shaped Earth arose when data was received from the U.S.'s artificial satellite [[Vanguard 1]] in 1958. It was found to vary in its long periodic orbit, with the Southern Hemisphere exhibiting higher gravitational attraction than the Northern Hemisphere. This indicated a flattening at the [[South Pole]] and a bulge of the same degree at the [[North Pole]], with the [[sea level]] increased about {{Convert|30|ft|abbr=on|order=flip|sigfig=1}} at the latter.<ref name=":0" /><ref name=":1" /><ref name="DMA" />{{Rp|page=9}} This theory implies the northern middle [[latitudes]] to be slightly flattened and the southern middle latitudes correspondingly bulged.<ref name="DMA" />{{Rp|page=9}} Potential factors involved in this aberration include [[tide]]s and [[Internal structure of Earth|subcrustal]] motion (e.g. [[plate tectonics]]).<ref name=":0">{{Cite book |last=Tyson |first=Neil deGrasse |author-link=Neil deGrasse Tyson |url=https://archive.org/details/NeilDeGrasseTysonDeathByBlackHoleAndOtherCosmicQuandaries/page/n61/mode/2up |title=Death By Black Hole: And Other Cosmic Quandaries |publisher=W. W. Norton |year=2007 |isbn=978-0-393-11378-5 |edition=1st |location=New York |pages=61}}</ref><ref name=":1">{{Cite book |last=Hawkins |first=Gerald S. |author-link=Gerald Hawkins |url=https://archive.org/details/splendorinsky00hawk/page/242/mode/2up |title=Splendor in the Sky |date=1969 |publisher=Harper & Row |location=New York |pages=242 |orig-date=1961}}</ref> [[John A. O'Keefe (astronomer)|John A. O'Keefe]] and co-authors are credited with the discovery that the Earth had a significant third degree [[zonal spherical harmonic]] in its [[Earth's gravity|gravitational field]] using Vanguard 1 satellite data.<ref>O’Keefe, J. A., Eckeis, A., and Squires, R. K. (1959). "Vanguard Measurements Give Pear-Shaped Component of Earth’s Figure". ''Science'', 129(3348), 565–566. {{doi|10.1126/science.129.3348.565}}.</ref> Based on further [[satellite geodesy]] data, [[Desmond King-Hele]] refined the estimate to a {{Convert|45|m|abbr=on}} difference between north and south polar radii, owing to a {{Convert|19|m|abbr=on}} "stem" rising in the North Pole and a {{Convert|26|m|abbr=on}} depression in the South Pole.<ref>{{cite journal | last1=King-Hele | first1=D. G. | last2=Cook | first2=G. E. | title=Refining the Earth's Pear Shape | journal=Nature | publisher=Springer Nature | volume=246 | issue=5428 | year=1973 | issn=0028-0836 | doi=10.1038/246086a0 | pages=86–88| bibcode=1973Natur.246...86K | s2cid=4260099 }}</ref><ref>{{cite journal|jstor=24926147 |title=The Shape of the Earth |last1=King-Hele |first1=Desmond |journal=Scientific American |date=1967 |volume=217 |issue=4 |pages=67–80 |doi=10.1038/scientificamerican1067-67 }}</ref> The polar asymmetry is about a thousand times smaller than the Earth's flattening and even smaller than its [[geoidal undulation]] in some regions.<ref>Günter Seeber (2008), ''[https://books.google.com/books?id=qZTS6OI9NGoC&dq=geoid%20j3%20pear&pg=PA517 Satellite Geodesy]'', Walter de Gruyter.</ref> === Geoid === {{Main|Geoid}} [[File:Earth_Gravitational_Model_1996.png|thumb|upright=1.5|Map of the undulation of the geoid in meters (based on the [[EGM96]] gravity model and the [[WGS84]] reference ellipsoid).]] Modern geodesy tends to retain the ellipsoid of revolution as a [[reference ellipsoid]] and treat triaxiality and pear shape as a part of the [[geoid]] figure: they are represented by the spherical harmonic coefficients <math>C_{22},S_{22}</math> and <math>C_{30}</math>, respectively, corresponding to degree and order numbers 2.2 for the triaxiality and 3.0 for the pear shape. It was stated earlier that measurements are made on the apparent or topographic surface of the Earth and it has just been explained that computations are performed on an ellipsoid. One other surface is involved in geodetic measurement: the geoid. In geodetic surveying, the computation of the [[geodetic coordinates]] of points is commonly performed on a [[reference ellipsoid]] closely approximating the size and shape of the Earth in the area of the survey. The actual measurements made on the surface of the Earth with certain instruments are however referred to the geoid. The ellipsoid is a mathematically defined regular surface with specific dimensions. The geoid, on the other hand, coincides with that surface to which the oceans would conform over the entire Earth if free to adjust to the combined effect of the Earth's mass attraction ([[gravitation]]) and the centrifugal force of the [[Earth's rotation]]. As a result of the uneven distribution of the Earth's mass, the geoidal surface is irregular and, since the ellipsoid is a regular surface, the separations between the two, referred to as [[geoid undulation]]s, geoid heights, or geoid separations, will be irregular as well. The geoid is a surface along which the gravity [[Equipotential|potential is equal]] everywhere and to which the direction of gravity is always perpendicular. The latter is particularly important because optical instruments containing gravity-reference leveling devices are commonly used to make geodetic measurements. When properly adjusted, the vertical axis of the instrument coincides with the direction of gravity and is, therefore, perpendicular to the geoid. The angle between the [[plumb line]] which is perpendicular to the geoid (sometimes called "the vertical") and the perpendicular to the ellipsoid (sometimes called "the ellipsoidal normal") is defined as the [[vertical deflection|deflection of the vertical]]. It has two components: an east–west and a north–south component.<ref name=DMA/> === Local approximations === Simpler local approximations are possible. ==== Local tangent plane ==== [[File:ECEF ENU Longitude Latitude relationships.svg|thumb|[[Local tangent plane]].]] The [[local tangent plane]] is appropriate for analysis across small distances. ==== Osculating sphere ==== [[File:Referenzellipsoid im x-z-Schnitt mit Kruemmungskreis.svg|thumb|left|Ellipsoid and osculating sphere]] The best local spherical approximation to the ellipsoid in the vicinity of a given point is the ''Earth's [[osculating circle|osculating]] sphere''. Its radius equals [[Earth radius#Gaussian radius of curvature|Earth's Gaussian radius of curvature]], and its radial direction coincides with the [[geodetic normal]] direction. The center of the osculating sphere is offset from the center of the ellipsoid, but is at the [[center of curvature]] for the given point on the ellipsoid surface. This concept aids the interpretation of terrestrial and planetary [[radio occultation]] [[refraction]] measurements and in some navigation and surveillance applications.<ref name="Williams">{{cite conference |title=On Loran-C Time-Difference to Co-ordinate Converters |first1=Paul |last1=Williams |first2=David |last2=Last |conference=International Loran Association (ILA) – 32nd Annual Convention and Technical Symposium |location=Boulder, Colorado |date=November 3–7, 2003 |citeseerx=10.1.1.594.6212 |url=https://loran.org/proceedings/Meeting2003/Session9/WmsLastILA03TD2LL.pdf |conference-url=https://loran.org/proceedings/Meeting2003/ProceedingsIndex.htm}}</ref><ref name="Razin">{{cite journal |title=Explicit (Noniterative) Loran Solution |first=Sheldon |last=Razin |journal=Navigation: Journal of the Institute of Navigation |volume=14 |issue=3 |date=Fall 1967 |pages=265–269 |doi=10.1002/j.2161-4296.1967.tb02208.x }}</ref> {{clear|left}} == Earth rotation and Earth's interior == {{main|Clairaut's theorem}} {{Further|Structure of Earth|Theoretical gravity}} Determining the exact figure of the Earth is not only a geometric task of geodesy, but also has [[geophysical]] considerations. According to theoretical arguments by Newton, [[Leonhard Euler]], and others, a body having a uniform density of 5,515&nbsp;kg/m{{sup|3}} that rotates like the Earth should have a flattening of 1:229. This can be concluded without any information about the composition of [[Earth's interior]].<ref> {{cite journal | title = Euler and the Flattening of the Earth | last = Heine | first = George | journal = Math Horizons | publisher = Mathematical Association of America | year = 2013 | volume = 21 | issue = 1 | pages = 25–29 | doi = 10.4169/mathhorizons.21.1.25 | s2cid = 126412032 | url = http://digitaleditions.walsworthprintgroup.com/publication/?i=172875&article_id=1491473&view=articleBrowser&ver=html5#{%22issue_id%22:172875,%22view%22:%22articleBrowser%22,%22article_id%22:%221491473%22} }}</ref> However, the measured flattening is 1:298.25, which is closer to a sphere and a strong argument that [[structure of the Earth|Earth's core]] is extremely compact. Therefore, the [[density]] must be a function of the depth, ranging from 2,600&nbsp;kg/m{{sup|3}} at the surface (rock density of [[granite]], etc.), up to 13,000&nbsp;kg/m{{sup|3}} within the inner core.<ref name=PREM>{{citation |last1=Dziewonski |first1=A. M. |last2=Anderson |first2=D. L. |title=Preliminary reference Earth model |journal=Physics of the Earth and Planetary Interiors |year= 1981 |volume=25 |issue=4 |pages=297–356 |url=https://www.cfa.harvard.edu/~lzeng/papers/PREM.pdf|doi=10.1016/0031-9201(81)90046-7 |issn=0031-9201 |bibcode=1981PEPI...25..297D}}</ref> == Global and regional gravity field == {{main|Gravity of Earth}} Also with implications for the physical exploration of the Earth's interior is the [[gravitational field]], which is the net effect of gravitation (due to mass attraction) and centrifugal force (due to rotation). It can be measured very accurately at the surface and remotely by satellites. True [[vertical direction|vertical]] generally does not correspond to theoretical vertical ([[deflection (physics)|deflection]] ranges up to 50") because [[topography]] and all ''geological masses'' disturb the gravitational field. Therefore, the gross structure of the [[Earth's crust]] and mantle can be determined by geodetic-geophysical models of the subsurface. == See also == * [[Clairaut's theorem]] * [[EGM96]] * [[Gravity formula]] * {{section link|Horizon|Distance|Curvature}} * [[Meridian arc]] ; History: * [[Pierre Bouguer]] *[[Earth's circumference#History]] *[[Earth's radius#History]] *[[Flat Earth]] * [[Friedrich Robert Helmert]] *[[History of geodesy]] *[[History of the metre]] *[[Meridian arc#History]] *[[Seconds pendulum]] == References == {{Reflist}} ;Attribution {{source-attribution|{{cite report | author = Defense Mapping Agency | date = 1983 | title = Geodesy for the Layman | url = https://apps.dtic.mil/sti/citations/ADA142764 | publisher = United States Air Force }}}} == Further reading == * [[Guy Bomford]], ''[[Geodesy (book)|Geodesy]]'', Oxford 1952 and 1980. * Guy Bomford, ''Determination of the European geoid by means of [[vertical deflection]]s''. Rpt of Comm. 14, [[IUGG]] 10th Gen. Ass., Rome 1954. * [[Karl Ledersteger]] and [[Gottfried Gerstbach]], ''Die horizontale [[Isostasy|Isostasie]] / Das isostatische Geoid 31. Ordnung''. Geowissenschaftliche Mitteilungen Band 5, [[TU Wien]] 1975. * [[Helmut Moritz]] and [[Bernhard Hofmann-Wellenhof|Bernhard Hofmann]], ''Physical Geodesy''. [[Springer Science+Business Media|Springer]], Wien & New York 2005. * ''Geodesy for the Layman'', [[Defense Mapping Agency]], St. Louis, 1983. == External links == {{Commons category|Curvature of the Earth}} *[https://web.archive.org/web/20051023083444/http://www.pcigeomatics.com/cgi-bin/pcihlp/PROJ%7CEARTH+MODELS%7CELLIPSOIDS%7CELLIPSOID+CODES Reference Ellipsoids (PCI Geomatics)] *[http://www.scanex.ru Reference Ellipsoids (ScanEx)] *[http://www.nasa.gov/centers/goddard/earthandsun/earthshape.html Changes in Earth shape due to climate changes] {{Webarchive|url=https://web.archive.org/web/20090122063915/http://www.nasa.gov/centers/goddard/earthandsun/earthshape.html |date=22 January 2009 }} *[http://www.josleys.com/show_gallery.php?galid=313 Jos Leys "The shape of Planet Earth"] {{Authority control}} [[Category:Earth]] [[Category:Geodesy]] [[Category:Geophysics]] </textarea><div class="templatesUsed"><div class="mw-templatesUsedExplanation"><p><span id="templatesused">Pages 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