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Parallel (geometry)

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	Topics Parallel (geometry) Topic Home Parallelism is a term in geometryGeometry Overview Geometry arose as the field of knowledge dealing with spatial relationships.... and in everyday life that refers to a property in Euclidean spaceEuclidean space Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "plane Euclidean geometry", wh... of two or more lineLine (mathematics) A line, or straight line, can be described as an infinitely thin, infinitely long, perfectly straight curve.... s or planesPlane (mathematics) In mathematics, a plane is a fundamental two-dimensional object.... , or a combination of these. The existence and properties of parallel lines are the basis of EuclidEuclid Overview Euclid , a Greek mathematician, who lived in Alexandria, Hellenistic Egypt, almost certainly during the reign of Ptolemy I... 's parallel postulateParallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate since it is the fifth postulate in Euclid's ... . Two lines parallel would be denoted as . Euclidean parallelism Given straight lines l and m, the following descriptions of line m equivalently define it as parallel to line l in Euclidean spaceEuclidean space Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "plane Euclidean geometry", wh... : Every point on line m is located exactly the same minimum distance from line l ('equidistant lines', not including the degenerate case where m = l). Line m is on the same plane as line l but does not intersect l (even assuming that lines extend to infinityInfinity he word infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts which arise i... in either direction). Lines m and l are both intersected by a third straight line (a transversalTransversal line In geometry, a transversal line is a line that passes through two or more other coplanar lines at different points, especial... ) in the same plane, and the corresponding angles of intersection with the transversal are equal. In other words, parallel lines must be located in the same plane, and parallel planes must be located in the same three-dimensional space. A parallel combination of a line and a plane may be located in the same three-dimensional space. Lines parallel to each other have the same gradient. Compare to perpendicularPerpendicular In geometry, two lines are considered perpendicular if one falls on the other in such a way as to create two equal angles.... . Construction The three definitions above lead to three different methods of construction of parallel lines. image:Par-equi.png\|Definition 1: Line m has everywhere the same distance to line l. image:Par-para.png\|Definition 2: Take a random line through a that intersects l in x. Move point x to infinity. image:Par-perp.png\|Definition 3: Both l and m share a transversal line through a that intersect them at 90°. Another definition of parallel line that's often used is that two lines are parallel if they do not intersect, though this definition applies only in the 2-dimensional plane. Another easy way is to remember that a parallel line is a line that has an equal distance with the opposite line. Extension to non-Euclidean geometry In Euclidean geometryEuclidean geometry Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria.... it is more common to talk about geodesicGeodesic In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".... s than (straight) lines. A geodesic is the path that a particle follows if no force is applied to it. In non-Euclidean geometry the above three definitions are not equivalent: only the second one is useful in other geometries. In general, equidistant lines are not geodesics so the equidistant definition cannot be used. In the Euclidean plane, when two geodesics (straight lines) are intersected with the same angles by a transversal geodesic (see image), every (non-parallel) geodesic intersects them with the same angles. In both the hyperbolic and spherical plane, this is not the case. E.g. geodesics sharing a common perpendicular only do so at one point (hyperbolic space) or at two (antipodal) points (spherical space). In general geometry it is useful to distinguish the three definitions above as three different types of lines, respectively equidistant lines, parallel geodesics and geodesics sharing a common perpendicular. While in Euclidean geometry two geodesics can either intersect or be parallel, in general and in hyperbolic space in particular there are three possibilities. Two geodesics can be either: intersecting: they intersect in a common point in the plane parallel: they do not intersect in the plane, but do in the limit to infinity ultra parallel: they do not even intersect in the limit to infinity In the literature ultra parallel geodesics are often called parallel. Geodesics intersecting at infinity are then called limit geodesics. Spherical In the spherical planeSpherical geometry Spherical geometry is the geometry of the two-dimensional surface of a sphere.... , all geodesics are great circles. Great circles divide the sphere in two equal hemispheresSphere A sphere is a perfectly symmetrical geometrical object.... and all great circles intersect each other. By the above definitions, there are no parallel geodesics to a given geodesic, all geodesics intersect. Equidistant lines on the sphere are called parallels of latitude in analog to latitudeLatitude Latitude, usually denoted symbolically by the Greek letter f , gives the location of a place on Earth north or south of the ... lines on a globe. These lines are not geodesics. An object traveling along such a line has to accelerate away from the geodesic it is equidistant to avoid intersecting with it. When embedded in Euclidean space a dimensionDimension In common usage, a dimension is a parameter or measurement required to define the characteristics of an object—i.e.... higher, parallels of latitude can be generated by the intersection of the sphere with a plane parallel to a plane through the center. Hyperbolic In the hyperbolic planeHyperbolic geometry Hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is rejected.... , there are two lines through a given point that intersect a given line in the limit to infinity. While in Euclidean geometry a geodesic intersects its parallels in both directions in the limit to infinity, in hyperbolic geometry both directions have their own line of parallelism. When visualized on a plane a geodesic is said to have a left handed parallel and a right handed parallel through a given point. The angle the parallel lines make with the perpendicular from that point to the given line is called the angle of parallelism. The angle of parallelism depends on the distance of the point to the line with respect to the curvatureCurvature Curvature refers to a number of loosely related concepts in different areas of geometry.... of the space. The angle is also present in the Euclidean case, there it is always 90° so the left and right handed parallels coincideCoincident Overview Coincident is a geometric term that pertains to the relationship between two vectors.... . The parallel lines divide the set of geodesics through the point in two sets: intersecting geodesics that intersect the given line in the hyperbolic plane, and ultra parallel geodesics that do not intersect even in the limit to infinity (in either direction). In the Euclidean limit the latter set is empty.