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42-587: Coordinates : 48°59′00″N 113°51′00″W / 48.98333°N 113.85000°W / 48.98333; -113.85000 Wilson Range [REDACTED] [REDACTED] Wilson Range Highest point Elevation 7,119 ft (2,170 m) Coordinates 48°59′00″N 113°51′00″W / 48.98333°N 113.85000°W / 48.98333; -113.85000 Geography Country United States State Montana The Wilson Range , el. 7,119 feet (2,170 m),
84-491: A 1 , … , a n ) {\displaystyle \left(a_{1},\ldots ,a_{n}\right)} may be identified with the ( surjective ) function with domain and with codomain that is defined at i ∈ domain F = { 1 , … , n } {\displaystyle i\in \operatorname {domain} F=\left\{1,\ldots ,n\right\}} by That is, F {\displaystyle F}
126-543: 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 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
168-487: A quaternion can be represented as a 4‑tuple, an octonion can be represented as an 8‑tuple, and a sedenion can be represented as a 16‑tuple. Although these uses treat ‑uple as the suffix, the original suffix was ‑ple as in "triple" (three-fold) or "decuple" (ten‑fold). This originates from medieval Latin plus (meaning "more") related to Greek ‑πλοῦς, which replaced the classical and late antique ‑plex (meaning "folded"), as in "duplex". The general rule for
210-638: A recurrence starting from ordered pairs ; indeed, an n -tuple can be identified with the ordered pair of its ( n − 1) first elements and its n th element. In computer science , tuples come in many forms. Most typed functional programming languages implement tuples directly as product types , tightly associated with algebraic data types , pattern matching , and destructuring assignment . Many programming languages offer an alternative to tuples, known as record types , featuring unordered elements accessed by label. A few programming languages combine ordered tuple product types and unordered record types into
252-411: A tuple is a finite sequence or ordered list of numbers or, more generally, mathematical objects , which are called the elements of the tuple. An n -tuple is a tuple of n elements, where n is a non-negative integer . There is only one 0-tuple, called the empty tuple . A 1-tuple and a 2-tuple are commonly called a singleton and an ordered pair , respectively. The term "infinite tuple"
294-679: 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 the longitude of the Royal Observatory in Greenwich , England as the zero-reference line. The Dominican Republic voted against
336-416: 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 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
378-538: 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 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,
420-431: A multiset and, in some non-English literature, variations with repetition . The number of n -tuples of an m -set is m . This follows from the combinatorial rule of product . If S is a finite set of cardinality m , this number is the cardinality of the n -fold Cartesian power S × S × ⋯ × S . Tuples are elements of this product set. In type theory , commonly used in programming languages ,
462-700: A national cartographical organization include the North American Datum , 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
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#1732765209877504-872: 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 ), 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
546-563: A single construct, as in C structs and Haskell records. Relational databases may formally identify their rows (records) as tuples . Tuples also occur in relational algebra ; when programming the semantic web with the Resource Description Framework (RDF); in linguistics ; and in philosophy . The term originated as an abstraction of the sequence: single, couple/double, triple, quadruple, quintuple, sextuple, septuple, octuple, ..., n ‑tuple, ..., where
588-426: A tuple has a product type ; this fixes not only the length, but also the underlying types of each component. Formally: and the projections are term constructors: The tuple with labeled elements used in the relational model has a record type . Both of these types can be defined as simple extensions of the simply typed lambda calculus . The notion of a tuple in type theory and that in set theory are related in
630-503: 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 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
672-586: Is where Earth's equatorial radius a {\displaystyle a} equals 6,378,137 m and tan β = b 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 }\,\!}
714-418: Is a spherical or geodetic coordinate system for measuring and communicating positions directly on Earth as latitude and longitude . It 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
756-820: Is a small mountain range in the northeast corner of Glacier National Park (U.S.) in Glacier County, Montana that parallels the international boundary with Canada . The range was named for Lt. Charles William Wilson (1836–1905) who was secretary to the British Boundary Commission (1858–1862). See also [ edit ] List of mountain ranges in Montana Mountains and mountain ranges of Glacier National Park (U.S.) Notes [ edit ] ^ "Wilson Range" . Geographic Names Information System . United States Geological Survey , United States Department of
798-408: Is as nested ordered pairs . This approach assumes that the notion of ordered pair has already been defined. This definition can be applied recursively to the ( n − 1) -tuple: Thus, for example: A variant of this definition starts "peeling off" elements from the other end: This definition can be applied recursively: Thus, for example: Using Kuratowski's representation for an ordered pair ,
840-701: Is known as the reduced (or parametric) latitude ). Aside from rounding, this 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. Tuple In mathematics ,
882-544: Is not cartesian because the measurements are angles and are not on a planar surface. A full GCS specification, such as those listed in 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
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#1732765209877924-480: Is occasionally used for "infinite sequences" . Tuples are usually written by listing the elements within parentheses " ( ) " and separated by commas; for example, (2, 7, 4, 1, 7) denotes a 5-tuple. Other types of brackets are sometimes used, although they may have a different meaning. An n -tuple can be formally defined as the image of a function that has the set of the n first natural numbers as its domain . Tuples may be also defined from ordered pairs by
966-753: 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 the French Institut national de l'information géographique et forestière —continue to use other meridians for internal purposes. The prime meridian determines
1008-469: Is the function defined by in which case the equality necessarily holds. Functions are commonly identified with their graphs , which is a certain set of ordered pairs. Indeed, many authors use graphs as the definition of a function. Using this definition of "function", the above function F {\displaystyle F} can be defined as: Another way of modeling tuples in Set Theory
1050-405: 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 a datum transformation such as a Helmert transformation , although in certain situations
1092-494: 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 the 1st or 2nd century, Marinus of Tyre compiled an extensive gazetteer and mathematically plotted world map using coordinates measured east from
1134-512: The Equator, one latitudinal second measures 30.715 m , one latitudinal minute is 1843 m and one latitudinal degree 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
1176-632: The Interior . ^ Holterman, Jack (1985). Place Names of Glacier/Waterton National Parks . West Glacier, Montana: Glacier Natural History Society. ISBN 0-916792-02-1 . Retrieved from " https://en.wikipedia.org/w/index.php?title=Wilson_Range&oldid=802828317 " Categories : Mountain ranges of Montana Landforms of Glacier County, Montana Hidden categories: Pages using gadget WikiMiniAtlas Coordinates on Wikidata Geographic coordinate system A geographic coordinate system ( GCS )
1218-464: 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 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 ,
1260-459: The following way: If we consider the natural model of a type theory, and use the Scott brackets to indicate the semantic interpretation, then the model consists of some sets S 1 , S 2 , … , S n {\displaystyle S_{1},S_{2},\ldots ,S_{n}} (note: the use of italics here that distinguishes sets from types) such that: and
1302-458: The identity of two n -tuples is Thus a tuple has properties that distinguish it from a set : There are several definitions of tuples that give them the properties described in the previous section. The 0 {\displaystyle 0} -tuple may be identified as the empty function . For n ≥ 1 , {\displaystyle n\geq 1,} the n {\displaystyle n} -tuple (
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1344-415: The length in meters of a degree of longitude can be calculated as (Those coefficients can be improved, but as they stand 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
1386-481: 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) the center of the Earth. Lines joining points of the same latitude trace circles on
1428-523: 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 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
1470-453: The prefixes are taken from the Latin names of the numerals. The unique 0-tuple is called the null tuple or empty tuple . A 1‑tuple is called a single (or singleton ), a 2‑tuple is called an ordered pair or couple , and a 3‑tuple is called a triple (or triplet ). The number n can be any nonnegative integer . For example, a complex number can be represented as a 2‑tuple of reals,
1512-535: 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 the International Date Line , which diverges from it in several places for political and convenience reasons, including between far eastern Russia and
1554-430: 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 the location has moved, but because the reference system used to measure it has shifted. Because any spatial reference system or map projection
1596-664: 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 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
1638-447: The second definition above can be reformulated in terms of pure set theory : In this formulation: In discrete mathematics , especially combinatorics and finite probability theory , n -tuples arise in the context of various counting problems and are treated more informally as ordered lists of length n . n -tuples whose entries come from a set of m elements are also called arrangements with repetition , permutations of
1680-486: 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 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
1722-464: 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 a point on Earth's surface
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1764-445: The width per minute and second, divide by 60 and 3600, respectively): where Earth's average meridional radius M r {\displaystyle \textstyle {M_{r}}\,\!} 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 }
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