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Parallel (geometry)
 
 
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Parallelism is a term in geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
 and in everyday life that refers to a property in Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
 of two or more line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
s or planes
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
, or a combination of these. The existence and properties of parallel lines are the basis of Euclid
Euclid

Euclid , floruit 300 BC, also known as Euclid of Alexandria, was a Greek mathematics and is often referred to as the Father of Geometry. He was active in Alexandria during the reign of Ptolemy I ....
's parallel postulate
Parallel postulate

In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in what is now called Euclidean geometry....
. Two lines parallel would be denoted as . The lines that do not intersect or meet are called parallel lines. For example, opposite edges of a ruler, rail lines, etc. are called parallel lines.

n straight lines l and m, the following descriptions of line m equivalently define it as parallel to line l in Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
:

  1. Every point on line m is located exactly the same minimum distance from line l (equidistant lines).
  2. Line m is on the same plane as line l but does not intersect l (even assuming that lines extend to infinity
    Infinity

    Infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts – usually linked to the idea of "without end" – which arise in philosophy, mathematics, and theology....
     in either direction).
  3. Lines m and l are both intersected by a third straight line (a transversal
    Transversal line

    In geometry, a transversal line is a Line that passes through two or more other coplanar lines at different Point .In Euclidean geometry if lines a and b are parallel, and line t intersects lines a and b, then corresponding angles formed by line t and the parallel lines are congruent....
    ) in the same plane, and the corresponding angles of intersection with the transversal are equal.






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    Parallelism is a term in geometry
    Geometry

    Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
     and in everyday life that refers to a property in Euclidean space
    Euclidean space

    Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
     of two or more line
    Line (mathematics)

    In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
    s or planes
    Plane (mathematics)

    In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
    , or a combination of these. The existence and properties of parallel lines are the basis of Euclid
    Euclid

    Euclid , floruit 300 BC, also known as Euclid of Alexandria, was a Greek mathematics and is often referred to as the Father of Geometry. He was active in Alexandria during the reign of Ptolemy I ....
    's parallel postulate
    Parallel postulate

    In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in what is now called Euclidean geometry....
    . Two lines parallel would be denoted as . The lines that do not intersect or meet are called parallel lines. For example, opposite edges of a ruler, rail lines, etc. are called parallel lines.

    Euclidean parallelism

    Parallel Lines
    Given straight lines l and m, the following descriptions of line m equivalently define it as parallel to line l in Euclidean space
    Euclidean space

    Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
    :

    1. Every point on line m is located exactly the same minimum distance from line l (equidistant lines).
    2. Line m is on the same plane as line l but does not intersect l (even assuming that lines extend to infinity
      Infinity

      Infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts – usually linked to the idea of "without end" – which arise in philosophy, mathematics, and theology....
       in either direction).
    3. Lines m and l are both intersected by a third straight line (a transversal
      Transversal line

      In geometry, a transversal line is a Line that passes through two or more other coplanar lines at different Point .In Euclidean geometry if lines a and b are parallel, and line t intersects lines a and b, then corresponding angles formed by line t and the parallel lines are congruent....
      ) in the same plane, and the corresponding angles of intersection with the transversal are equal. (This is equivalent to Euclid
      Euclid

      Euclid , floruit 300 BC, also known as Euclid of Alexandria, was a Greek mathematics and is often referred to as the Father of Geometry. He was active in Alexandria during the reign of Ptolemy I ....
      's parallel postulate
      Parallel postulate

      In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in what is now called Euclidean geometry....
      .)


    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 perpendicular
    Perpendicular

    In geometry, two line or plane , are considered perpendicular to each other if they form congruence adjacent angles angles . The term may be used as a noun or adjective....
    .

    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.

    Distance between two parallel lines

    Because a parallel line is a line that has an equal distance with the opposite line, there is a unique distance between the two parallel lines. Given the equations of two non-vertical parallel lines:



    the distance between the two lines can be formulated by following formula:



    Extension to non-Euclidean geometry

    In Euclidean geometry
    Euclidean geometry

    Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
     it is more common to talk about geodesic
    Geodesic

    In mathematics, a geodesic [jee-uh-des-ik, -dee-sik] is a generalization of the notion of a "Line " to "manifolds".In presence of a Metric , geodesics are defined to be the shortest path between points on the space....
    s than (straight) lines. A geodesic is the path that a particle follows if no force is applied to it. In non-Euclidean geometry (spherical
    Elliptic geometry

    Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a Point p outside L, there exists no line Parallel to L passing through p....
     or hyperbolic
    Hyperbolic geometry

    In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced. The parallel postulate in Euclidean geometry is equivalent to the statement that, in two dimensional space, for any given line l and point P not on l, there is exactly one line through P th...
    ) 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:

    1. intersecting: they intersect in a common point in the plane
    2. parallel: they do not intersect in the plane, but do in the limit to infinity
    3. 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 plane
    Spherical geometry

    Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a non-Euclidean geometry. Two practical applications of the principles of spherical geometry are navigation and astronomy....
    , all geodesics are great circles. Great circles divide the sphere in two equal hemispheres
    Sphere

    A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
     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 latitude
    Latitude

    Latitude, usually denoted symbolically by the Greek letter phi gives the location of a place on Earth north or south of the equator. Lines of Latitude are the horizontal lines shown running east-to-west on maps ....
     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 dimension
    Dimension

    In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
     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 plane
    Hyperbolic geometry

    In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced. The parallel postulate in Euclidean geometry is equivalent to the statement that, in two dimensional space, for any given line l and point P not on l, there is exactly one line through P th...
    , 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 curvature
    Curvature

    In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line , but this is defined in different ways depending on the context....
     of the space. The angle is also present in the Euclidean case, there it is always 90° so the left and right-handed parallels coincide
    Coincident

    Coincident is a geometry term that pertains to the relationship between two Vector . Vectors consist of a "magnitude" and a "direction". Vectors can be said to be coincident when their direction is the same though the magnitude may be different; that is to say, they lie one on top of the other....
    . 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.

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