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59-638: Sur Lípez is one of sixteen provinces in the Potosí Department. Also, the southwesternmost point of Bolivia is located here, at 22° 49' 41.016" South , 67° 52' 35.004" West , at an elevation of approximately 5,400 m on the northeastern slope of the Licancabur volcano. It is bordered by the Nor Lípez Province in the north and northwest, Enrique Baldivieso Province in the west, the Republic of Chile in

118-404: A and semi-minor axis c , therefore e may be identified as the eccentricity . (See ellipse .) A prolate spheroid with c > a has surface area The prolate spheroid is generated by rotation about the z -axis of an ellipse with semi-major axis c and semi-minor axis a ; therefore, e may again be identified as the eccentricity . (See ellipse .) These formulas are identical in

177-481: 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

236-560: 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

295-458: 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 the definitions of latitude and longitude. In

354-404: A massive body in a close orbit. The most extreme example is Jupiter's moon Io , which becomes slightly more or less prolate in its orbit due to a slight eccentricity, causing intense volcanism . The major axis of the prolate spheroid does not run through the satellite's poles in this case, but through the two points on its equator directly facing toward and away from the primary. This combines with

413-443: 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

472-549: 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

531-425: Is Both of these curvatures are always positive, so that every point on a spheroid is elliptic. The aspect ratio of an oblate spheroid/ellipse, c  : a , is the ratio of the polar to equatorial lengths, while the flattening (also called oblateness ) f , is the ratio of the equatorial-polar length difference to the equatorial length: The first eccentricity (usually simply eccentricity, as above)

590-406: Is If A = 2 a is the equatorial diameter, and C = 2 c is the polar diameter, the volume is Let a spheroid be parameterized as where β is the reduced latitude or parametric latitude , λ is the longitude , and − ⁠ π / 2 ⁠ < β < + ⁠ π / 2 ⁠ and −π < λ < +π . Then, the spheroid's Gaussian curvature is and its mean curvature

649-487: Is Uturunku at 6,008 m (19,711 ft). Other mountains are listed below: The tourist circuit La Ruta de las Joyas Altoandinas passes through the spectacular geography of this area. The province comprises three municipalities are further subdivided into cantons . The population increased from 4,158 (1992 census) to 4,905 inhabitants (2001 census), an increase of 18%. 99.4% of the population have no access to electricity, and 90% have no sanitary facilities. 69% of

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708-471: 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 the equator. Latitude and longitude are used together as a coordinate pair to specify

767-399: Is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters . A spheroid has circular symmetry . If the ellipse is rotated about its major axis , the result is a prolate spheroid , elongated like a rugby ball . The American football is similar but has a pointier end than a spheroid could. If

826-411: 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

885-484: 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

944-453: 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

1003-401: Is given by setting a = b : The semi-axis a is the equatorial radius of the spheroid, and c is the distance from centre to pole along the symmetry axis. There are two possible cases: The case of a = c reduces to a sphere. An oblate spheroid with c < a has surface area The oblate spheroid is generated by rotation about the z -axis of an ellipse with semi-major axis

1062-462: 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

1121-478: Is often used instead of flattening. It is defined by: The relations between eccentricity and flattening are: All modern geodetic ellipsoids are defined by the semi-major axis plus either the semi-minor axis (giving the aspect ratio), the flattening, or the first eccentricity. While these definitions are mathematically interchangeable, real-world calculations must lose some precision. To avoid confusion, an ellipsoidal definition considers its own values to be exact in

1180-451: 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

1239-538: Is the approximate shape of the ball in several sports, such as in the rugby ball . Several moons of the Solar System approximate prolate spheroids in shape, though they are actually triaxial ellipsoids . Examples are Saturn 's satellites Mimas , Enceladus , and Tethys and Uranus ' satellite Miranda . In contrast to being distorted into oblate spheroids via rapid rotation, celestial objects distort slightly into prolate spheroids via tidal forces when they orbit

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1298-421: 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

1357-505: 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

1416-974: The Maclaurin spheroid and the Jacobi ellipsoid . Spheroid is also a shape of archaeological artifacts. The oblate spheroid is the approximate shape of rotating planets and other celestial bodies , including Earth, Saturn , Jupiter , and the quickly spinning star Altair . Saturn is the most oblate planet in the Solar System , with a flattening of 0.09796. See planetary flattening and equatorial bulge for details. Enlightenment scientist Isaac Newton , working from Jean Richer 's pendulum experiments and Christiaan Huygens 's theories for their interpretation, reasoned that Jupiter and Earth are oblate spheroids owing to their centrifugal force . Earth's diverse cartographic and geodetic systems are based on reference ellipsoids , all of which are oblate. The prolate spheroid

1475-400: The actinide and lanthanide elements are shaped like prolate spheroids. In anatomy, near-spheroid organs such as testis may be measured by their long and short axes . Many submarines have a shape which can be described as prolate spheroid. For a spheroid having uniform density, the moment of inertia is that of an ellipsoid with an additional axis of symmetry. Given a description of

1534-571: 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

1593-409: The Earth is often approximated by an oblate spheroid, known as the reference ellipsoid , instead of a sphere. The current World Geodetic System model uses a spheroid whose radius is 6,378.137 km (3,963.191 mi) at the Equator and 6,356.752 km (3,949.903 mi) at the poles . The word spheroid originally meant "an approximately spherical body", admitting irregularities even beyond

1652-447: 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

1711-664: 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

1770-567: 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

1829-425: 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

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1888-454: The bi- or tri-axial ellipsoidal shape; that is how the term is used in some older papers on geodesy (for example, referring to truncated spherical harmonic expansions of the Earth's gravity geopotential model ). The equation of a tri-axial ellipsoid centred at the origin with semi-axes a , b and c aligned along the coordinate axes is The equation of a spheroid with z as the symmetry axis

1947-572: 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

2006-408: 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

2065-500: The ellipse is rotated about its minor axis , the result is an oblate spheroid , flattened like a lentil or a plain M&;M . If the generating ellipse is a circle, the result is a sphere . Due to the combined effects of gravity and rotation , the figure of the Earth (and of all planets ) is not quite a sphere, but instead is slightly flattened in the direction of its axis of rotation. For that reason, in cartography and geodesy

2124-402: 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 ) ,

2183-509: 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

2242-483: 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

2301-438: 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

2360-522: The form it gives. The most common shapes for the density distribution of protons and neutrons in an atomic nucleus are spherical , prolate, and oblate spheroidal, where the polar axis is assumed to be the spin axis (or direction of the spin angular momentum vector). Deformed nuclear shapes occur as a result of the competition between electromagnetic repulsion between protons, surface tension and quantum shell effects . Spheroids are common in 3D cell cultures . Rotating equilibrium spheroids include

2419-399: 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

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2478-408: The largest being Laguna Colorada , which is 10 km in diameter at an elevation of 4,278 m. Other lakes, such as Laguna Verde , Laguna Blanca and Laguna Celeste are also well known for their respectively green, white and blue colors. There is a geyser field called " Sol de Mañana " in southwestern Sur Lípez. The Cordillera de Lípez traverses the province. The highest mountain of the province

2537-451: 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

2596-538: 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

2655-572: 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: Ellipsoid of revolution A spheroid , also known as an ellipsoid of revolution or rotational ellipsoid ,

2714-458: 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

2773-454: The population is employed in agriculture, 4% in mining, 4% in industry, and 23% in general services. 86% of the population are Catholics, 9% are Protestants. The people are predominantly indigenous citizens of Quechua descent. The languages spoken in the province are mainly Spanish and Quechua. 22°04′S 67°07′W  /  22.067°S 67.117°W  / -22.067; -67.117 Latitude In geography , latitude

2832-503: 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

2891-466: 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

2950-417: 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

3009-515: 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

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3068-417: The sense that the formula for S oblate can be used to calculate the surface area of a prolate spheroid and vice versa. However, e then becomes imaginary and can no longer directly be identified with the eccentricity. Both of these results may be cast into many other forms using standard mathematical identities and relations between parameters of the ellipse. The volume inside a spheroid (of any kind)

3127-486: The smaller oblate distortion from the synchronous rotation to cause the body to become triaxial. The term is also used to describe the shape of some nebulae such as the Crab Nebula . Fresnel zones , used to analyze wave propagation and interference in space, are a series of concentric prolate spheroids with principal axes aligned along the direct line-of-sight between a transmitter and a receiver. The atomic nuclei of

3186-564: The southwest and south, the Republic of Argentina in the southeast and east, and Sud Chichas Province in the northeast. The province extends from the northeast to the southwest, at a length of 230 km and an average width of 100 km. In its southwestern part, the Quetena Grande Canton lies in the Eduardo Avaroa Andean Fauna National Reserve , the province has a couple of lakes and salt pans ,

3245-420: 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 )

3304-403: 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

3363-454: 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

3422-468: 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

3481-407: 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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