Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position The existence and properties of parallel lines are the basis of Euclid's parallel postulate. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive Two lines parallel would be denoted as ABC DEF.

Euclidean parallelism

As shown by the tick marks, lines a and b are parallel. We can prove this because the transversal t produces congruent angles. In Geometry, a Transversal line is a line that passes through two or more other Coplanar lines at different points.

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

1. 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).
2. Line m is on the same plane as line l but does not intersect l (even assuming that lines extend to infinity in either direction). Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness
3. Lines m and l are both intersected by a third straight line (a transversal) in the same plane, and the corresponding angles of intersection with the transversal are equal. In Geometry, a Transversal line is a line that passes through two or more other Coplanar lines at different points.

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. In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent

Construction

The three definitions above lead to three different methods of construction of parallel lines.

The problem: Draw a line through a parallel to l.

Definition 1: Line m has everywhere the same distance to line l.

Definition 2: Take a random line through a that intersects l in x. Move point x to infinity.

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 geometry it is more common to talk about geodesics than (straight) lines. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces A geodesic is the path that a particle follows if no force is applied to it. In non-Euclidean geometry (spherical or hyperbolic) the above three definitions are not equivalent: only the second one is useful in other geometries. Elliptic geometry (sometimes known as Riemannian geometry) is a Non-Euclidean geometry, in which given a line L and a point In In general, equidistant lines are not geodesics so the equidistant definition cannot be used. Distance is a numerical description of how far apart objects are 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

On the spherical plane there is no such thing as a parallel line. Line a is a great circle, the equivalent of a straight line in the spherical plane. A great circle is a Circle on the surface of a Sphere that has the same circumference as the sphere dividing the sphere into two equal Hemispheres. Line c is equidistant to line a but is not a great circle. It is a parallel of latitude. Line b is another geodesic which intersects a in two antipodal points. They share two common perpendiculars (one shown in blue).

In the spherical plane, all geodesics are great circles. Spherical geometry is the Geometry of the two- Dimensional surface of a Sphere. A great circle is a Circle on the surface of a Sphere that has the same circumference as the sphere dividing the sphere into two equal Hemispheres. Great circles divide the sphere in two equal hemispheres and all great circles intersect each other. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe 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 lines on a globe. Latitude, usually denoted symbolically by the Greek letter phi ( Φ) gives the location of a place on Earth (or other planetary body north or south of the 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 higher, parallels of latitude can be generated by the intersection of the sphere with a plane parallel to a plane through the center. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it

Hyperbolic

Intersecting, parallel and ultra parallel lines trough a with respect to l in the hyperbolic plane. The parallel lines appear to intersect l just off the image. This is an artifact of the visualisation. It is not possible to isometrically embed the hyperbolic plane in three dimension. In a real hyperbolic space the line will get closer to each other and 'touch' in infinity.

In the hyperbolic plane, there are two lines through a given point that intersect a given line in the limit to infinity. In 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 of the space. In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry The angle is also present in the Euclidean case, there it is always 90° so the left and right handed parallels coincide. 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.

• Constructing a parallel line through a given point with compass and straightedge