Longitude

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A graticule on the Earth as a sphere or an ellipsoid. The lines from pole to pole are lines of constant longitude, or meridians. The circles parallel to the Equator are circles of constant latitude, or parallels. The graticule shows the latitude and longitude of points on the surface. In this example, meridians are spaced at 6deg intervals and parallels at 4deg intervals. Division of the Earth into Gauss-Krueger zones - Globe.svg
A graticule on the Earth as a sphere or an ellipsoid. The lines from pole to pole are lines of constant longitude, or meridians. The circles parallel to the Equator are circles of constant latitude, or parallels. The graticule shows the latitude and longitude of points on the surface. In this example, meridians are spaced at 6° intervals and parallels at 4° intervals.

Longitude ( /ˈlɒnɪtjd/ , AU and UK also /ˈlɒŋɡɪ-/ ), [1] [2] is a geographic coordinate that specifies the eastwest position of a point on the Earth's surface, or the surface of a celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter lambda (λ). Meridians (lines running from pole to pole) connect points with the same longitude. By convention, one of these, the Prime Meridian, which passes through the Royal Observatory, Greenwich, England, was allocated the position of 0° longitude. The longitude of other places is measured as the angle east or west from the Prime Meridian, ranging from 0° at the Prime Meridian to +180° eastward and −180° westward. Specifically, it is the angle between a plane through the Prime Meridian and a plane through both poles and the location in question. (This forms a right-handed coordinate system with the z-axis (right hand thumb) pointing from the Earth's center toward the North Pole and the x-axis (right hand index finger) extending from the Earth's center through the Equator at the Prime Meridian.)

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A location's northsouth position along a meridian is given by its latitude, which is approximately the angle between the local vertical and the equatorial plane.

If the Earth were perfectly spherical and radially homogeneous, then the longitude at a point would be equal to the angle between a vertical north–south plane through that point and the plane of the Greenwich meridian. Everywhere on Earth the vertical north–south plane would contain the Earth's axis. But the Earth is not radially homogeneous and has rugged terrain, which affect gravity and so can shift the vertical plane away from the Earth's axis. The vertical north–south plane still intersects the plane of the Greenwich meridian at some angle; that angle is the astronomical longitude, calculated from star observations. The longitude shown on maps and GPS devices is the angle between the Greenwich plane and a not-quite-vertical plane through the point; the not-quite-vertical plane is perpendicular to the surface of the spheroid chosen to approximate the Earth's sea-level surface, rather than perpendicular to the sea-level surface itself.

History

Amerigo Vespucci's observations: the origins of the "lunar distance method" Longitude Vespucci.png
Amerigo Vespucci's observations: the origins of the "lunar distance method"

The measurement of longitude is important both to cartography and for ocean navigation. Mariners and explorers for most of history struggled to determine longitude. Finding a method of determining longitude took centuries, resulting in the history of longitude recording requiring the effort of some of the greatest scientific minds.

Latitude was calculated by observing with quadrant or astrolabe the altitude of the sun or of charted stars above the horizon, but longitude is harder.

Amerigo Vespucci appears as being the first European to proffer a solution in writing, after devoting a great deal of time and energy studying the problem during his sojourns in the New World:

As to longitude, I declare that I found so much difficulty in determining it that I was put to great pains to ascertain the east-west distance I had covered. The final result of my labours was that I found nothing better to do than to watch for and take observations at night of the conjunction of one planet with another, and especially of the conjunction of the moon with the other planets, because the moon is swifter in her course than any other planet. I compared my observations with an almanac. After I had made experiments many nights, one night, the twenty-third of August 1499, there was a conjunction of the moon with Mars, which according to the almanac was to occur at midnight or a half hour before. I found that...at midnight Mars's position was three and a half degrees to the east. [3]

John Harrison solved the greatest problem of his day. John Harrison Uhrmacher.jpg
John Harrison solved the greatest problem of his day.

Anyway, the genuinity and integrity of Vespucci's writings is deemed questionable (they appear as having been polluted and corrupted by forgers).(see reference) By comparing the timing of the observed positions of the moon and Mars with the expected timing of their occurrence when seen from Europe (Florence or Nurnberg), Vespucci would actually have been able to deduce his longitude (measured in hours). However, for (t)his method to provide usable results, three conditions should be satisfied: First, it requires waiting for a specific expected astronomical event to occur (in this case, Mars passing through the same right ascension as the moon), and its occurrence to be accurately timed in an astronomical almanac. Second, one needs to know the precise local time, something difficult to ascertain for Vespucci, when travelling expecially. Third, it requires a stable viewing platform, rendering the technique useless on the rolling deck of a ship at sea. See Lunar distance (navigation). Due to none of such conditions being fullfilled when Vespucci took his measures (except for the third one, if he was on the ground), there was no possibility for him to accurately esteem his longitude. Notheless he must be credited with having envisioned a correct method to measure longitude (in hours, easily transformed into degrees: 1 hour = 15 degrees). Such a method was to be named the "Lunar distance method" and it became usable by mariners much later, when lunar positions started to be known and published in an accurate enough Almanac: the one issued by astronomer Nevil Maskelyne (1732-1811).see reference

In 1612 Galileo Galilei demonstrated that with sufficiently accurate knowledge of the orbits of the moons of Jupiter one could use their positions as a universal clock and this would make possible the determination of longitude, but the method he devised was impracticable for navigators on ships because of their instability. [5] In 1714 the British government passed the Longitude Act which offered large financial rewards to the first person to demonstrate a practical method for determining the longitude of a ship at sea. These rewards motivated many to search for a solution.

Drawing of Earth with longitudes but without latitudes. Longitude (PSF).png
Drawing of Earth with longitudes but without latitudes.

John Harrison, a self-educated English clockmaker, invented the marine chronometer, the key piece in solving the problem of accurately establishing longitude at sea, thus revolutionising and extending the possibility of safe long distance sea travel. [4] A French expedition under Charles-François-César Le Tellier de Montmirail performed the first measurement of longitude aboard Aurore in 1767. [6] Though the Board of Longitude rewarded John Harrison for his marine chronometer in 1773, chronometers remained very expensive and the lunar distance method continued to be used for decades. Finally, the combination of the availability of marine chronometers and wireless telegraph time signals put an end to the use of lunars in the 20th century.

Unlike latitude, which has the equator as a natural starting position, there is no natural starting position for longitude. Therefore, a reference meridian had to be chosen. It was a popular practice to use a nation's capital as the starting point, but other locations were also used. While British cartographers had long used the Greenwich meridian in London, other references were used elsewhere, including El Hierro, Rome, Copenhagen, Jerusalem, Saint Petersburg, Pisa, Paris (see the article Paris meridian ), Philadelphia, and Washington D.C. In 1884 the International Meridian Conference adopted the Greenwich meridian as the universal Prime Meridian or zero point of longitude. [7]

Noting and calculating longitude

Longitude is given as an angular measurement ranging from 0° at the Prime Meridian to +180° eastward and −180° westward. The Greek letter λ (lambda), [8] [9] is used to denote the location of a place on Earth east or west of the Prime Meridian.

Each degree of longitude is sub-divided into 60 minutes, each of which is divided into 60 seconds. A longitude is thus specified in sexagesimal notation as 23° 27′ 30″ E. For higher precision, the seconds are specified with a decimal fraction. An alternative representation uses degrees and minutes, where parts of a minute are expressed in decimal notation with a fraction, thus: 23° 27.5′ E. Degrees may also be expressed as a decimal fraction: 23.45833° E. For calculations, the angular measure may be converted to radians, so longitude may also be expressed in this manner as a signed fraction of π (pi), or an unsigned fraction of 2π.

For calculations, the West/East suffix is replaced by a negative sign in the western hemisphere. The preferred convention—that East is positive—is consistent with a right-handed Cartesian coordinate system, with the North Pole up. A specific longitude may then be combined with a specific latitude (usually positive in the northern hemisphere) to give a precise position on the Earth's surface. Confusingly, the convention of negative for East is also sometimes seen, most commonly in the United States; the Earth System Research Laboratory notes that, while it differs from the international standard, it "make(s) coordinate entry less awkward" for applications confined to the Western Hemisphere. [10]

There is no other physical principle determining longitude directly but with time. Longitude at a point may be determined by calculating the time difference between that at its location and Coordinated Universal Time (UTC). Since there are 24 hours in a day and 360 degrees in a circle, the sun moves across the sky at a rate of 15 degrees per hour (360° ÷ 24 hours = 15° per hour). So if the time zone a person is in is three hours ahead of UTC then that person is near 45° longitude (3 hours × 15° per hour = 45°). The word near is used because the point might not be at the center of the time zone; also the time zones are defined politically, so their centers and boundaries often do not lie on meridians at multiples of 15°. In order to perform this calculation, however, a person needs to have a chronometer (watch) set to UTC and needs to determine local time by solar or astronomical observation. The details are more complex than described here: see the articles on Universal Time and on the equation of time for more details.

Singularity and discontinuity of longitude

Note that the longitude is singular at the Poles and calculations that are sufficiently accurate for other positions may be inaccurate at or near the Poles. Also the discontinuity at the ±180° meridian must be handled with care in calculations. An example is a calculation of east displacement by subtracting two longitudes, which gives the wrong answer if the two positions are on either side of this meridian. To avoid these complexities, consider replacing latitude and longitude with another horizontal position representation in calculation.

Plate movement and longitude

The Earth's tectonic plates move relative to one another in different directions at speeds on the order of 50 to 100 mm (2.0 to 3.9 in) per year. [11] So points on the Earth's surface on different plates are always in motion relative to one another. For example, the longitudinal difference between a point on the Equator in Uganda, on the African Plate, and a point on the Equator in Ecuador, on the South American Plate, is increasing by about 0.0014 arcseconds per year. These tectonic movements likewise affect latitude.

If a global reference frame (such as WGS84, for example) is used, the longitude of a place on the surface will change from year to year. To minimize this change, when dealing just with points on a single plate, a different reference frame can be used, whose coordinates are fixed to a particular plate, such as "NAD83" for North America or "ETRS89" for Europe.

Length of a degree of longitude

The length of a degree of longitude (east-west distance) depends only on the radius of a circle of latitude. For a sphere of radius a that radius at latitude φ is a cos φ, and the length of a one-degree (or π/180 radian) arc along a circle of latitude is

φΔ1
lat
Δ1
long
110.574 km111.320 km
15°110.649 km107.551 km
30°110.852 km96.486 km
45°111.133 km78.847 km
60°111.412 km55.800 km
75°111.618 km28.902 km
90°111.694 km0.000 km
Length of one degree (black), minute (blue) and second (red) of latitude and longitude in metric (upper half) and imperial units (lower half) at a given latitude (vertical axis) in WGS84. For example, the green arrows show that Donetsk (green circle) at 48degN has a Dlong of 74.63 km/deg (1.244 km/min, 20.73 m/sec etc) and a Dlat of 111.2 km/deg (1.853 km/min, 30.89 m/sec etc). WGS84 angle to distance conversion.svg
Length of one degree (black), minute (blue) and second (red) of latitude and longitude in metric (upper half) and imperial units (lower half) at a given latitude (vertical axis) in WGS84. For example, the green arrows show that Donetsk (green circle) at 48°N has a Δlong of 74.63 km/° (1.244 km/min, 20.73 m/sec etc) and a Δlat of 111.2 km/° (1.853 km/min, 30.89 m/sec etc).

When the Earth is modelled by an ellipsoid this arc length becomes [12] [13]

where e, the eccentricity of the ellipsoid, is related to the major and minor axes (the equatorial and polar radii respectively) by

An alternative formula is

Cos φ decreases from 1 at the equator to 0 at the poles, which measures how circles of latitude shrink from the equator to a point at the pole, so the length of a degree of longitude decreases likewise. This contrasts with the small (1%) increase in the length of a degree of latitude (north-south distance), equator to pole. The table shows both for the WGS84 ellipsoid with a = 6378137.0 m and b = 6356752.3142 m. Note that the distance between two points 1 degree apart on the same circle of latitude, measured along that circle of latitude, is slightly more than the shortest (geodesic) distance between those points (unless on the equator, where these are equal); the difference is less than 0.6 m (2 ft).

A geographical mile is defined to be the length of one minute of arc along the equator (one equatorial minute of longitude), therefore a degree of longitude along the equator is exactly 60 geographical miles or 111.3 kilometers, as there are 60 minutes in a degree. The length of 1 minute of longitude along the equator is 1 geographical mile or 1.855 km or 1.153 miles, while the length of 1 second of it is 0.016 geographical mile or 30.916 m or 101.43 feet.

Longitude on bodies other than Earth

Planetary co-ordinate systems are defined relative to their mean axis of rotation and various definitions of longitude depending on the body. The longitude systems of most of those bodies with observable rigid surfaces have been defined by references to a surface feature such as a crater. The north pole is that pole of rotation that lies on the north side of the invariable plane of the solar system (near the ecliptic). The location of the prime meridian as well as the position of the body's north pole on the celestial sphere may vary with time due to precession of the axis of rotation of the planet (or satellite). If the position angle of the body's prime meridian increases with time, the body has a direct (or prograde) rotation; otherwise the rotation is said to be retrograde.

In the absence of other information, the axis of rotation is assumed to be normal to the mean orbital plane; Mercury and most of the satellites are in this category. For many of the satellites, it is assumed that the rotation rate is equal to the mean orbital period. In the case of the giant planets, since their surface features are constantly changing and moving at various rates, the rotation of their magnetic fields is used as a reference instead. In the case of the Sun, even this criterion fails (because its magnetosphere is very complex and does not really rotate in a steady fashion), and an agreed-upon value for the rotation of its equator is used instead.

For planetographic longitude, west longitudes (i.e., longitudes measured positively to the west) are used when the rotation is prograde, and east longitudes (i.e., longitudes measured positively to the east) when the rotation is retrograde. In simpler terms, imagine a distant, non-orbiting observer viewing a planet as it rotates. Also suppose that this observer is within the plane of the planet's equator. A point on the Equator that passes directly in front of this observer later in time has a higher planetographic longitude than a point that did so earlier in time.

However, planetocentric longitude is always measured positively to the east, regardless of which way the planet rotates. East is defined as the counter-clockwise direction around the planet, as seen from above its north pole, and the north pole is whichever pole more closely aligns with the Earth's north pole. Longitudes traditionally have been written using "E" or "W" instead of "+" or "−" to indicate this polarity. For example, −91°, 91°W, +269° and 269°E all mean the same thing.

The reference surfaces for some planets (such as Earth and Mars) are ellipsoids of revolution for which the equatorial radius is larger than the polar radius, such that they are oblate spheroids. Smaller bodies (Io, Mimas, etc.) tend to be better approximated by triaxial ellipsoids; however, triaxial ellipsoids would render many computations more complicated, especially those related to map projections. Many projections would lose their elegant and popular properties. For this reason spherical reference surfaces are frequently used in mapping programs.

The modern standard for maps of Mars (since about 2002) is to use planetocentric coordinates. Guided by the works of historical astronomers, Merton E. Davies established the meridian of Mars at Airy-0 crater. [14] [15] For Mercury, the only other planet with a solid surface visible from Earth, a thermocentric coordinate is used: the prime meridian runs through the point on the equator where the planet is hottest (due to the planet's rotation and orbit, the sun briefly retrogrades at noon at this point during perihelion, giving it more sun). By convention, this meridian is defined as exactly twenty degrees of longitude east of Hun Kal. [16] [17] [18]

Tidally-locked bodies have a natural reference longitude passing through the point nearest to their parent body: 0° the center of the primary-facing hemisphere, 90° the center of the leading hemisphere, 180° the center of the anti-primary hemisphere, and 270° the center of the trailing hemisphere. [19] However, libration due to non-circular orbits or axial tilts causes this point to move around any fixed point on the celestial body like an analemma.

See also

Related Research Articles

Latitude geographic coordinate specifying north–south position

In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the geodetic latitude as defined below. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six auxiliary latitudes which are used in special applications.

Mercator projection Map projection for navigational use that distorts areas far from the equator

The Mercator projection is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because of its unique property of representing any course of constant bearing as a straight segment. Such a course, known as a rhumb or, mathematically, a loxodrome, is preferred by navigators because the ship can sail in a constant compass direction to reach its destination, eliminating difficult and error-prone course corrections. Linear scale is constant on the Mercator in every direction around any point, thus preserving the angles and the shapes of small objects and fulfilling the conditions of a conformal map projection. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation is very small near the equator, but accelerates with latitude to become infinite at the poles. So, for example, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator, such as Central Africa.

Spherical coordinate system 3-dimensional coordinate system

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

Geographic coordinate system Coordinate system

A geographic coordinate system is a coordinate system that enables every location on Earth to be specified by a set of numbers, letters or symbols. The coordinates are often chosen such that one of the numbers represents a vertical position and two or three of the numbers represent a horizontal position; alternatively, a geographic position may be expressed in a combined three-dimensional Cartesian vector. A common choice of coordinates is latitude, longitude and elevation. To specify a location on a plane requires a map projection.

Hour angle unit of angle

In astronomy and celestial navigation, the hour angle is one of the coordinates used in the equatorial coordinate system to give the direction of a point on the celestial sphere. The hour angle of a point is the angle between two planes: one containing Earth's axis and the zenith, and the other containing Earth's axis and the given point.

Prime meridian A line of longitude, at which longitude is defined to be 0°

A prime meridian is the meridian in a geographic coordinate system at which longitude is defined to be 0°. Together, a prime meridian and its anti-meridian form a great circle. This great circle divides a spheroid into two hemispheres. If one uses directions of East and West from a defined prime meridian, then they can be called the Eastern Hemisphere and the Western Hemisphere.

Earth radius mean distance from the Earths center to its surface

Earth radius is the distance from the center of Earth to a point on its surface. Its value ranges from 6,378 km (3,963 mi) at the equator to 6,357 km (3,950 mi) at a pole. A nominal Earth radius is sometimes used as a unit of measurement in astronomy and geophysics, denoted in astronomy by the symbol R. In other contexts, it is denoted or sometimes .

Rhumb line arc crossing all meridians of longitude at the same angle

In navigation, a rhumb line, rhumb, or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true or magnetic north.

Great-circle distance shortest distance between two points along the surface of a sphere

The great-circle distance or orthodromic distance is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere. The distance between two points in Euclidean space is the length of a straight line between them, but on the sphere there are no straight lines. In spaces with curvature, straight lines are replaced by geodesics. Geodesics on the sphere are circles on the sphere whose centers coincide with the center of the sphere, and are called great circles.

Prime meridian (Greenwich) meridian

The future prime meridian based at the Royal Observatory, Greenwich, in London, England, was established by Sir George Airy in 1851. By 1884, over two-thirds of all ships and tonnage used it as the reference meridian on their charts and maps. In October of that year, at the behest of US President Chester A. Arthur, 41 delegates from 25 nations met in Washington, D.C., United States, for the International Meridian Conference. This conference selected the meridian passing through Greenwich as the official prime meridian due to its popularity. However, France abstained from the vote, and French maps continued to use the Paris meridian for several decades. In the 18th century, London lexicographer Malachy Postlethwayt published his African maps showing the "Meridian of London" intersecting the Equator a few degrees west of the later meridian and Accra, Ghana.

Reference ellipsoid Ellipsoid that approximates the figure of the Earth

In geodesy, a reference ellipsoid is a mathematically defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body. 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.

Meridian (geography) line between the poles with the same longitude

A (geographic) meridian is the half of an imaginary great circle on the Earth's surface, terminated by the North Pole and the South Pole, connecting points of equal longitude, as measured in angular degrees east or west of the Prime Meridian. The position of a point along the meridian is given by that longitude and its latitude, measured in angular degrees north or south of the Equator. Each meridian is perpendicular to all circles of latitude. Each is also the same length, being half of a great circle on the Earth's surface and therefore measuring 20,003.93 km.

Transverse Mercator projection The transverse Mercator projection is the transverse aspect of the standard (or Normal) Mercator projection

The transverse Mercator map projection is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the UTM. When paired with a suitable geodetic datum, the transverse Mercator delivers high accuracy in zones less than a few degrees in east-west extent.

Scale (map) Ratio of distance on a map to the corresponding distance on the ground

The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways.

Equirectangular projection map projection that maps meridians and parallels to vertical and horizontal straight lines, respectively, producing a rectangular grid

The equirectangular projection is a simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100. The projection maps meridians to vertical straight lines of constant spacing, and circles of latitude to horizontal straight lines of constant spacing. The projection is neither equal area nor conformal. Because of the distortions introduced by this projection, it has little use in navigation or cadastral mapping and finds its main use in thematic mapping. In particular, the plate carrée has become a standard for global raster datasets, such as Celestia and NASA World Wind, because of the particularly simple relationship between the position of an image pixel on the map and its corresponding geographic location on Earth.

Universal Transverse Mercator coordinate system coordinate system

The Universal Transverse Mercator (UTM) is a system for assigning coordinates to locations on the surface of the Earth. Like the traditional method of latitude and longitude, it is a horizontal position representation, which means it ignores altitude and treats the earth as a perfect ellipsoid. However, it differs from global latitude/longitude in that it divides earth into 60 zones and projects each to the plane as a basis for its coordinates. Specifying a location means specifying the zone and the x, y coordinate in that plane. The projection from spheroid to a UTM zone is some parameterization of the transverse Mercator projection. The parameters vary by nation or region or mapping system.

Longitude by chronometer is a method, in navigation, of determining longitude using a marine chronometer, which was developed by John Harrison during the first half of the eighteenth century. It is an astronomical method of calculating the longitude at which a position line, drawn from a sight by sextant of any celestial body, crosses the observer's assumed latitude. In order to calculate the position line, the time of the sight must be known so that the celestial position i.e. the Greenwich Hour Angle and Declination, of the observed celestial body is known. All that can be derived from a single sight is a single position line, which can be achieved at any time during daylight when both the sea horizon and the sun are visible. To achieve a fix, more than one celestial body and the sea horizon must be visible. This is usually only possible at dawn and dusk.

The n-vector representation is a three-parameter non-singular representation well-suited for replacing latitude and longitude as horizontal position representation in mathematical calculations and computer algorithms.

Great ellipse

A great ellipse is an ellipse passing through two points on a spheroid and having the same center as that of the spheroid. Equivalently, it is an ellipse on the surface of a spheroid and centered on the origin, or the curve formed by intersecting the spheroid by a plane through its center. For points that are separated by less than about a quarter of the circumference of the earth, about , the length of the great ellipse connecting the points is close to the geodesic distance. The great ellipse therefore is sometimes proposed as a suitable route for marine navigation. The great ellipse is special case of an earth section path.

Geographical distance Distance measured along the surface of the earth

Geographical distance is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude. This distance is an element in solving the second (inverse) geodetic problem.

References

  1. "Definition of LONGITUDE". www.merriam-webster.com. Merriam-Webster . Retrieved 14 March 2018.
  2. Oxford English Dictionary
  3. Vespucci, Amerigo. "Letter from Seville to Lorenzo di Pier Francesco de' Medici, 1500." Pohl, Frederick J. Amerigo Vespucci: Pilot Major. New York: Columbia University Press, 1945. 76–90. Page 80.
  4. 1 2 "Longitude clock comes alive". BBC. March 11, 2002.
  5. Denny, Mark (2012), The Science of Navigation: From Dead Reckoning to GPS, Johns Hopkins University Press, p. 105, ISBN   9781421405605, in 1610, Galileo thought he might win the Spanish longitude prize with his idea of measuring time by observing the moons of Jupiter ... The trouble with the method was in making accurate measurements of the four moons while on the deck of a moving ship at sea. This problem proved intractable, and the method was therefore not adopted.
  6. "MONOGRAPHIE DE L'AURORE - Corvette -1766". Ancre. Retrieved 2019-12-05.
  7. "International Conference Held at Washington for the Purpose of Fixing a Prime Meridian and a Universal Day. October, 1884. Protocols of the proceedings". Project Gutenberg. 1884. Retrieved 30 November 2012.
  8. "Coordinate Conversion". colorado.edu. Archived from the original on 29 September 2009. Retrieved 14 March 2018.
  9. "λ = Longitude east of Greenwich (for longitude west of Greenwich, use a minus sign)."
    John P. Snyder, Map Projections, A Working Manual , USGS Professional Paper 1395, page ix
  10. NOAA ESRL Sunrise/Sunset Calculator (deprecated). Earth System Research Laboratory . Retrieved October 18, 2019.
  11. Read HH, Watson Janet (1975). Introduction to Geology. New York: Halsted. pp. 13–15.
  12. Osborne, Peter (2013). "Chapter 5: The geometry of the ellipsoid". The Mercator Projections: The Normal and Transverse Mercator Projections on the Sphere and the Ellipsoid with Full Derivations of all Formulae (PDF). Edinburgh. doi:10.5281/zenodo.35392. Archived from the original (PDF) on 2016-05-09. Retrieved 2016-01-24.
  13. Rapp, Richard H. (April 1991). "Chapter 3: Properties of the Ellipsoid". Geometric Geodesy Part I. Columbus, Ohio.: Department of Geodetic Science and Surveying, Ohio State University. hdl:1811/24333.
  14. Where is zero degrees longitude on Mars? – Copyright 2000 – 2010 © European Space Agency. All rights reserved.
  15. Davies, M. E., and R. A. Berg, "Preliminary Control Net of Mars,"Journal of Geophysical Research, Vol. 76, No. 2, pps. 373-393, January 10, 1971.
  16. Davies, M. E., "Surface Coordinates and Cartography of Mercury," Journal of Geophysical Research, Vol. 80, No. 17, June 10, 1975.
  17. Archinal, Brent A.; A'Hearn, Michael F.; Bowell, Edward L.; Conrad, Albert R.; et al. (2010). "Report of the IAU Working Group on Cartographic Coordinates and Rotational Elements: 2009". Celestial Mechanics and Dynamical Astronomy. 109 (2): 101–135. Bibcode:2011CeMDA.109..101A. doi:10.1007/s10569-010-9320-4. ISSN   0923-2958.
  18. "USGS Astrogeology: Rotation and pole position for the Sun and planets (IAU WGCCRE)". Archived from the original on October 24, 2011. Retrieved October 22, 2009.
  19. First map of extraterrestrial planet – Center of Astrophysics.