48-535: Parallels may refer to: Circle of latitude (also parallels), an abstract east–west small circle connecting all locations around Earth at a given latitude Parallels (company) , a software company based in Bellevue, Washington Parallels Desktop for Mac , software providing hardware virtualization for Macintosh computers with Intel processors Parallels Server for Mac , server-side desktop virtualization product built for
96-710: A circle of latitude is given by its longitude . Circles of latitude are unlike circles of longitude, which are all great circles with the centre of Earth in the middle, as the circles of latitude get smaller as the distance from the Equator increases. Their length can be calculated by a common sine or cosine function. For example, the 60th parallel north or south is half as long as the Equator (disregarding Earth's minor flattening by 0.335%), stemming from cos ( 60 ∘ ) = 0.5 {\displaystyle \cos(60^{\circ })=0.5} . On
144-495: A map, the circles of latitude may or may not be parallel, and their spacing may vary, depending on which projection is used to map the surface of the Earth onto a plane. On an equirectangular projection , centered on the equator, the circles of latitude are horizontal, parallel, and equally spaced. On other cylindrical and pseudocylindrical projections, the circles of latitude are horizontal and parallel, but may be spaced unevenly to give
192-431: A pair of lines in congruent corresponding angles then all transversals must do so. Again, a new axiom is needed to justify this statement. The three properties above lead to three different methods of construction of parallel lines. Because parallel lines in a Euclidean plane are equidistant there is a unique distance between the two parallel lines. Given the equations of two non-vertical, non-horizontal parallel lines,
240-489: A pair of straight lines in a plane which do not meet appears as Definition 23 in Book I of Euclid's Elements . Alternative definitions were discussed by other Greeks, often as part of an attempt to prove the parallel postulate . Proclus attributes a definition of parallel lines as equidistant lines to Posidonius and quotes Geminus in a similar vein. Simplicius also mentions Posidonius' definition as well as its modification by
288-413: A plane q in three-dimensional space, the line not lying in that plane, are parallel if and only if they do not intersect. Equivalently, they are parallel if and only if the distance from a point P on line m to the nearest point in plane q is independent of the location of P on line m . Similar to the fact that parallel lines must be located in the same plane, parallel planes must be situated in
336-595: Is ∥ {\displaystyle \parallel } . For example, A B ∥ C D {\displaystyle AB\parallel CD} indicates that line AB is parallel to line CD . In the Unicode character set, the "parallel" and "not parallel" signs have codepoints U+2225 (∥) and U+2226 (∦), respectively. In addition, U+22D5 (⋕) represents the relation "equal and parallel to". Given parallel straight lines l and m in Euclidean space ,
384-536: Is different from Wikidata All article disambiguation pages All disambiguation pages Circle of latitude A circle of latitude or line of latitude on Earth is an abstract east – west small circle connecting all locations around Earth (ignoring elevation ) at a given latitude coordinate line . Circles of latitude are often called parallels because they are parallel to each other; that is, planes that contain any of these circles never intersect each other. A location's position along
432-505: Is drawn as a "line on a map", which was made in massive scale during the 1884 Berlin Conference , regarding huge parts of the African continent. North American nations and states have also mostly been created by straight lines, which are often parts of circles of latitudes. For instance, the northern border of Colorado is at 41° N while the southern border is at 37° N . Roughly half
480-516: Is equal to the Earth's axial tilt. By definition, the positions of the Tropic of Cancer , Tropic of Capricorn , Arctic Circle and Antarctic Circle all depend on the tilt of the Earth's axis relative to the plane of its orbit around the Sun (the "obliquity of the ecliptic"). If the Earth were "upright" (its axis at right angles to the orbital plane) there would be no Arctic, Antarctic, or Tropical circles: at
528-486: Is evidently a symmetric relation . According to Euclid's tenets, parallelism is not a reflexive relation and thus fails to be an equivalence relation . Nevertheless, in affine geometry a pencil of parallel lines is taken as an equivalence class in the set of lines where parallelism is an equivalence relation. To this end, Emil Artin (1957) adopted a definition of parallelism where two lines are parallel if they have all or none of their points in common. Then
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#1732851771183576-496: Is replaced by the more general concept of a geodesic , a curve which is locally straight with respect to the metric (definition of distance) on a Riemannian manifold , a surface (or higher-dimensional space) which may itself be curved. In general relativity , particles not under the influence of external forces follow geodesics in spacetime , a four-dimensional manifold with 3 spatial dimensions and 1 time dimension. In non-Euclidean geometry ( elliptic or hyperbolic geometry )
624-463: Is the longest circle of latitude and is the only circle of latitude which also is a great circle. As such, it is perpendicular to all meridians. There are 89 integral (whole degree ) circles of latitude between the Equator and the poles in each hemisphere , but these can be divided into more precise measurements of latitude, and are often represented as a decimal degree (e.g. 34.637° N) or with minutes and seconds (e.g. 22°14'26" S). On
672-441: Is the treatment of parallel lines. These reform texts were not without their critics and one of them, Charles Dodgson (a.k.a. Lewis Carroll ), wrote a play, Euclid and His Modern Rivals , in which these texts are lambasted. One of the early reform textbooks was James Maurice Wilson's Elementary Geometry of 1868. Wilson based his definition of parallel lines on the primitive notion of direction . According to Wilhelm Killing
720-641: The Mercator projection or on the Gall-Peters projection , a circle of latitude is perpendicular to all meridians . On the ellipsoid or on spherical projection, all circles of latitude are rhumb lines , except the Equator. The latitude of the circle is approximately the angle between the Equator and the circle, with the angle's vertex at Earth's centre. The Equator is at 0°, and the North Pole and South Pole are at 90° north and 90° south, respectively. The Equator
768-446: The same direction , but are not parts of the same straight line, are called parallel lines ." Wilson (1868 , p. 12) Augustus De Morgan reviewed this text and declared it a failure, primarily on the basis of this definition and the way Wilson used it to prove things about parallel lines. Dodgson also devotes a large section of his play (Act II, Scene VI § 1) to denouncing Wilson's treatment of parallels. Wilson edited this concept out of
816-534: The American science fiction television series "Parallels", song by As I Lay Dying from the album The Powerless Rise , 2010 "Parallels", song by Fit for a King from the album Descendants , 2011 "Parallels", song by Misery Signals from the album Controller , 2008 "Parallels", song by Yes from the album Going for the One See also [ edit ] Parallel (disambiguation) Topics referred to by
864-535: The Equator, mark the divisions between the five principal geographical zones . The equator is the circle that is equidistant from the North Pole and South Pole . It divides the Earth into the Northern Hemisphere and the Southern Hemisphere . Of the parallels or circles of latitude, it is the longest, and the only ' great circle ' (a circle on the surface of the Earth, centered on Earth's center). All
912-456: The Mac OS X Server platform Parallels Workstation , first commercial software product released by Parallels, Inc. Parallels (engineering) , rectangular blocks of metal which have faces ground or lapped to a precise surface finish Arts & entertainment [ edit ] British Blockade (also Parallels), a patience game or solitaire of the blockade family Parallels (album) ,
960-599: The Tropical Circles are drifting towards the equator (and the Polar Circles towards the poles) by 15 m per year, and the area of the Tropics , defined astronomically, is decreasing by 1,100 km (420 sq mi) per year. (However, the tropical belt as defined based on atmospheric conditions is expanding due to global warming . ) The Earth's axial tilt has additional shorter-term variations due to nutation , of which
1008-495: The Tropics and Polar Circles and also on the Equator. Short-term fluctuations over a matter of days do not directly affect the location of the extreme latitudes at which the Sun may appear directly overhead, or at which 24-hour day or night is possible, except when they actually occur at the time of the solstices. Rather, they cause a theoretical shifting of the parallels, that would occur if the given axis tilt were maintained throughout
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#17328517711831056-408: The distance between the two lines can be found by locating two points (one on each line) that lie on a common perpendicular to the parallel lines and calculating the distance between them. Since the lines have slope m , a common perpendicular would have slope −1/ m and we can take the line with equation y = − x / m as a common perpendicular. Solve the linear systems and to get the coordinates of
1104-558: The following properties are equivalent: Since these are equivalent properties, any one of them could be taken as the definition of parallel lines in Euclidean space, but the first and third properties involve measurement, and so, are "more complicated" than the second. Thus, the second property is the one usually chosen as the defining property of parallel lines in Euclidean geometry. The other properties are then consequences of Euclid's Parallel Postulate . The definition of parallel lines as
1152-502: The horizon for 24 hours (at the December and June Solstices respectively). The latitude of the polar circles is equal to 90° minus the Earth's axial tilt . The Tropic of Cancer and Tropic of Capricorn mark the northernmost and southernmost latitudes at which the Sun may be seen directly overhead at the June solstice and December solstice respectively. The latitude of the tropical circles
1200-413: The idea may be traced back to Leibniz . Wilson, without defining direction since it is a primitive, uses the term in other definitions such as his sixth definition, "Two straight lines that meet one another have different directions, and the difference of their directions is the angle between them." Wilson (1868 , p. 2) In definition 15 he introduces parallel lines in this way; "Straight lines which have
1248-526: The length of the border between the United States and Canada follows 49° N . There are five major circles of latitude, listed below from north to south. The position of the Equator is fixed (90 degrees from Earth's axis of rotation) but the latitudes of the other circles depend on the tilt of this axis relative to the plane of Earth's orbit, and so are not perfectly fixed. The values below are for 22 November 2024: These circles of latitude, excluding
1296-399: The main term, with a period of 18.6 years, has an amplitude of 9.2″ (corresponding to almost 300 m north and south). There are many smaller terms, resulting in varying daily shifts of some metres in any direction. Finally, the Earth's rotational axis is not exactly fixed in the Earth, but undergoes small fluctuations (on the order of 15 m) called polar motion , which have a small effect on
1344-627: The map useful characteristics. For instance, on a Mercator projection the circles of latitude are more widely spaced near the poles to preserve local scales and shapes, while on a Gall–Peters projection the circles of latitude are spaced more closely near the poles so that comparisons of area will be accurate. On most non-cylindrical and non-pseudocylindrical projections, the circles of latitude are neither straight nor parallel. Arcs of circles of latitude are sometimes used as boundaries between countries or regions where distinctive natural borders are lacking (such as in deserts), or when an artificial border
1392-433: The mean value of the tilt was 23° 26′ 21.406″ (according to IAU 2006, theory P03), the corresponding value being 23° 26′ 10.633" at noon of January 1st 2023 AD. The main long-term cycle causes the axial tilt to fluctuate between about 22.1° and 24.5° with a period of 41,000 years. Currently, the average value of the tilt is decreasing by about 0.468″ per year. As a result (approximately, and on average),
1440-481: The northern hemisphere because astronomic latitude can be roughly measured (to within a few tens of metres) by sighting the North Star . Normally the circles of latitude are defined at zero elevation . Elevation has an effect on a location with respect to the plane formed by a circle of latitude. Since (in the geodetic system ) altitude and depth are determined by the normal to the Earth's surface, locations sharing
1488-608: The other parallels are smaller and centered only on Earth's axis. The Arctic Circle is the southernmost latitude in the Northern Hemisphere at which the Sun can remain continuously above or below the horizon for 24 hours (at the June and December solstices respectively). Similarly, the Antarctic Circle marks the northernmost latitude in the Southern Hemisphere at which the Sun can remain continuously above or below
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1536-526: The philosopher Aganis. At the end of the nineteenth century, in England, Euclid's Elements was still the standard textbook in secondary schools. The traditional treatment of geometry was being pressured to change by the new developments in projective geometry and non-Euclidean geometry , so several new textbooks for the teaching of geometry were written at this time. A major difference between these reform texts, both between themselves and between them and Euclid,
1584-410: The points. The solutions to the linear systems are the points and These formulas still give the correct point coordinates even if the parallel lines are horizontal (i.e., m = 0). The distance between the points is which reduces to When the lines are given by the general form of the equation of a line (horizontal and vertical lines are included): their distance can be expressed as Two lines in
1632-485: The poles the Sun would always circle along the horizon, and at the equator the Sun would always rise due east, pass directly overhead, and set due west. The positions of the Tropical and Polar Circles are not fixed because the axial tilt changes slowly – a complex motion determined by the superimposition of many different cycles (some of which are described below) with short to very long periods. At noon of January 1st 2000 AD,
1680-404: The problem that the points that are found at a fixed given distance on one side of a straight line must be shown to form a straight line. This can not be proved and must be assumed to be true. The corresponding angles formed by a transversal property, used by W. D. Cooley in his 1860 text, The Elements of Geometry, simplified and explained requires a proof of the fact that if one transversal meets
1728-434: The same direction or opposite direction (not necessarily the same length). Parallel lines are the subject of Euclid 's parallel postulate . Parallelism is primarily a property of affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry , lines can have analogous properties that are referred to as parallelism. The parallel symbol
1776-446: The same three-dimensional space that do not intersect need not be parallel. Only if they are in a common plane are they called parallel; otherwise they are called skew lines . Two distinct lines l and m in three-dimensional space are parallel if and only if the distance from a point P on line m to the nearest point on line l is independent of the location of P on line m . This never holds for skew lines. A line m and
1824-400: The same three-dimensional space that never meet. Parallel curves are curves that do not touch each other or intersect and keep a fixed minimum distance. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called skew lines . Line segments and Euclidean vectors are parallel if they have
1872-514: The same latitude—but having different elevations (i.e., lying along this normal)—no longer lie within this plane. Rather, all points sharing the same latitude—but of varying elevation and longitude—occupy the surface of a truncated cone formed by the rotation of this normal around the Earth's axis of rotation. Parallel (geometry) In geometry , parallel lines are coplanar infinite straight lines that do not intersect at any point. Parallel planes are planes in
1920-653: The same plane can either be: In the literature ultra parallel geodesics are often called non-intersecting . Geodesics intersecting at infinity are called limiting parallel . As in the illustration through a point a not on line l there are two limiting parallel lines, one for each direction ideal point of line l. They separate the lines intersecting line l and those that are ultra parallel to line l . Ultra parallel lines have single common perpendicular ( ultraparallel theorem ), and diverge on both sides of this common perpendicular. In spherical geometry , all geodesics are great circles . Great circles divide
1968-415: The same term [REDACTED] This disambiguation page lists articles associated with the title Parallels . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Parallels&oldid=1217374647 " Category : Disambiguation pages Hidden categories: Short description
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2016-401: The same three-dimensional space and contain no point in common. Two distinct planes q and r are parallel if and only if the distance from a point P in plane q to the nearest point in plane r is independent of the location of P in plane q . This will never hold if the two planes are not in the same three-dimensional space. In non-Euclidean geometry , the concept of a straight line
2064-419: The sixth studio album by American progressive metal band Fates Warning Parallels (band) , a Canadian synthpop band from Toronto Parallels (film) , a 2015 American science-fiction adventure film Parallels (TV series) , a French science-fiction-mystery streaming television series for children and adolescents "Parallels" ( Star Trek: The Next Generation ) , the 11th episode of the seventh season of
2112-410: The sphere in two equal hemispheres and all great circles intersect each other. Thus, there are no parallel geodesics to a given geodesic, as all geodesics intersect. Equidistant curves on the sphere are called parallels of latitude analogous to the latitude lines on a globe. Parallels of latitude can be generated by the intersection of the sphere with a plane parallel to a plane through the center of
2160-506: The sphere. If l, m, n are three distinct lines, then l ∥ m ∧ m ∥ n ⟹ l ∥ n . {\displaystyle l\parallel m\ \land \ m\parallel n\ \implies \ l\parallel n.} In this case, parallelism is a transitive relation . However, in case l = n , the superimposed lines are not considered parallel in Euclidean geometry. The binary relation between parallel lines
2208-443: The third and higher editions of his text. Other properties, proposed by other reformers, used as replacements for the definition of parallel lines, did not fare much better. The main difficulty, as pointed out by Dodgson, was that to use them in this way required additional axioms to be added to the system. The equidistant line definition of Posidonius, expounded by Francis Cuthbertson in his 1874 text Euclidean Geometry suffers from
2256-582: The three Euclidean properties mentioned above are not equivalent and only the second one (Line m is in the same plane as line l but does not intersect l) is useful in non-Euclidean geometries, since it involves no measurements. In general geometry the three properties above give three different types of curves, equidistant curves , parallel geodesics and geodesics sharing a common perpendicular , respectively. While in Euclidean geometry two geodesics can either intersect or be parallel, in hyperbolic geometry, there are three possibilities. Two geodesics belonging to
2304-445: The year. These circles of latitude can be defined on other planets with axial inclinations relative to their orbital planes. Objects such as Pluto with tilt angles greater than 45 degrees will have the tropic circles closer to the poles and the polar circles closer to the equator. A number of sub-national and international borders were intended to be defined by, or are approximated by, parallels. Parallels make convenient borders in
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