In geometry, **parallel** lines are lines in a plane which do not meet; that is, two straight lines in a plane that do not intersect at any point are said to be parallel. Colloquially, curves that do not touch each other or intersect and keep a fixed minimum distance are said to be parallel. A line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are also said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel; otherwise they are called skew lines. Parallel planes are planes in the same three-dimensional space that never meet.

- Symbol
- Euclidean parallelism
- Two lines in a plane
- Two lines in three-dimensional space
- A line and a plane
- Two planes
- Extension to non-Euclidean geometry
- Hyperbolic geometry
- Spherical or elliptic geometry
- Reflexive variant
- See also
- Notes
- References
- Further reading
- External links

Parallel lines are the subject of Euclid's parallel postulate.^{ [1] } 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 is .^{ [2] }^{ [3] } For example, 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".^{ [4] }

The same symbol is used for a binary function in electrical engineering (the parallel operator). It is distinct from the double-vertical-line brackets that indicate a norm, as well as from the logical or operator (`||`

) in several programming languages.

Given parallel straight lines *l* and *m* in Euclidean space, the following properties are equivalent:

- Every point on line
*m*is located at exactly the same (minimum) distance from line*l*(*equidistant lines*). - Line
*m*is in the same plane as line*l*but does not intersect*l*(recall that lines extend to infinity in either direction). - When lines
*m*and*l*are both intersected by a third straight line (a transversal) in the same plane, the corresponding angles of intersection with the transversal are congruent.

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.^{ [5] } The other properties are then consequences of Euclid's Parallel Postulate. Another property that also involves measurement is that lines parallel to each other have the same gradient (slope).

The definition of parallel lines as a pair of straight lines in a plane which do not meet appears as Definition 23 in Book I of Euclid's Elements.^{ [6] } 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 the philosopher Aganis.^{ [6] }

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, is the treatment of parallel lines.^{ [7] } 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.^{ [8] }

One of the early reform textbooks was James Maurice Wilson's *Elementary Geometry* of 1868.^{ [9] } Wilson based his definition of parallel lines on the primitive notion of *direction*. According to Wilhelm Killing ^{ [10] } the idea may be traced back to Leibniz.^{ [11] } 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 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 the third and higher editions of his text.^{ [12] }

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 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.^{ [13] } 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 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^{ [14] } 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,

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 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 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 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 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, it is more common to talk about geodesics than (straight) lines. A geodesic is the shortest path between two points in a given geometry. In physics this may be interpreted as the path that a particle follows if no force is applied to it. In non-Euclidean geometry (elliptic or hyperbolic geometry) 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 ) since it involves no measurements is useful in non-Euclidean geometries. 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 the same plane can either be:

**intersecting**, if they intersect in a common point in the plane,**parallel**, if they do not intersect in the plane, but converge to a common limit point at infinity (ideal point), or**ultra parallel**, if they do not have a common limit point at infinity.

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 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 the sphere.

If *l, m, n* are three distinct lines, then

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 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.^{ [15] }^{ [16] }^{ [17] }

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.^{ [18] } Then a line *is* parallel to itself so that the reflexive and transitive properties belong to this type of parallelism, creating an equivalence relation on the set of lines. In the study of incidence geometry, this variant of parallelism is used in the affine plane.

- ↑ Although this postulate only refers to when lines meet, it is needed to prove the uniqueness of parallel lines in the sense of Playfair's axiom.
- ↑ Kersey (the elder), John (1673).
*Algebra*. Book IV. London. p. 177. - ↑ Cajori, Florian (1993) [September 1928]. "§ 184, § 359, § 368".
*A History of Mathematical Notations - Notations in Elementary Mathematics*.**1**(two volumes in one unaltered reprint ed.). Chicago, US: Open court publishing company. pp. 193, 402–403, 411–412. ISBN 0-486-67766-4. LCCN 93-29211 . Retrieved 2019-07-22.§359. […] ∥ for parallel occurs in Oughtred's

*Opuscula mathematica hactenus inedita*(1677) [p. 197], a posthumous work (§ 184) […] §368. Signs for parallel lines. […] when Recorde's sign of equality won its way upon the Continent, vertical lines came to be used for parallelism. We find ∥ for "parallel" in Kersey,[14] Caswell, Jones,[15] Wilson,[16] Emerson,[17] Kambly,[18] and the writers of the last fifty years who have been already quoted in connection with other pictographs. Before about 1875 it does not occur as often […] Hall and Stevens[1] use "par[1] or ∥" for parallel […] [14] John Kersey,*Algebra*(London, 1673), Book IV, p. 177. [15] W. Jones,*Synopsis palmarioum matheseos*(London, 1706). [16] John Wilson,*Trigonometry*(Edinburgh, 1714), characters explained. [17] W. Emerson,*Elements of Geometry*(London, 1763), p. 4. [18] L. Kambly ,*Die Elementar-Mathematik*, Part 2:*Planimetrie*, 43. edition (Breslau, 1876), p. 8. […] [1] H. S. Hall and F. H. Stevens,*Euclid's Elements*, Parts I and II (London, 1889), p. 10. […] - ↑ "Mathematical Operators – Unicode Consortium" (PDF). Retrieved 2013-04-21.
- ↑ Wylie Jr. 1964 , pp. 92—94
- 1 2 Heath 1956 , pp. 190–194
- ↑ Richards 1988 , Chap. 4: Euclid and the English Schoolchild. pp. 161–200
- ↑ Carroll, Lewis (2009) [1879],
*Euclid and His Modern Rivals*, Barnes & Noble, ISBN 978-1-4351-2348-9 - ↑ Wilson 1868
- ↑
*Einführung in die Grundlagen der Geometrie, I*, p. 5 - ↑ Heath 1956 , p. 194
- ↑ Richards 1988 , pp. 180–184
- ↑ Heath 1956 , p. 194
- ↑ Only the third is a straightedge and compass construction, the first two are infinitary processes (they require an "infinite number of steps".)
- ↑ H. S. M. Coxeter (1961)
*Introduction to Geometry*, p 192, John Wiley & Sons - ↑ Wanda Szmielew (1983)
*From Affine to Euclidean Geometry*, p 17, D. Reidel ISBN 90-277-1243-3 - ↑ Andy Liu (2011) "Is parallelism an equivalence relation?", The College Mathematics Journal 42(5):372
- ↑ Emil Artin (1957)
*Geometric Algebra*, page 52 via Internet Archive

A **circle** is a shape consisting of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.

**Euclidean space** is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-dimensional space and the *Euclidean plane*. It was introduced by the Ancient Greek mathematician Euclid of Alexandria, and the qualifier *Euclidean* is used to distinguish it from other spaces that were later discovered in physics and modern mathematics.

A **sphere** is a geometrical object in three-dimensional space that is the surface of a ball.

In mathematics, **non-Euclidean geometry** consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either relaxing the metric requirement, or replacing the parallel postulate with an alternative. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry.

In mathematics, a **plane** is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point, a line and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with one of a room's walls, infinitely extended, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.

**Elliptic geometry** is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point. Because of this, the elliptic geometry described in this article is sometimes referred to as *single elliptic geometry* whereas spherical geometry is sometimes referred to as *double elliptic geometry*.

In mathematics, **hyperbolic geometry** is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

In geometry, **inversive geometry** is the study of *inversion*, a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry become much more tractable when an inversion is applied.

In mathematics, a **hyperbolic space** is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature. It is hyperbolic geometry in more than 2 dimensions, and is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, and elliptic geometry that have a constant positive curvature.

In non-Euclidean geometry, the **Poincaré half-plane model** is the upper half-plane, denoted below as **H**, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.

In geometry, the notion of **line** or **straight line** was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, which are often described in terms of two points or referred to using a single letter.

In hyperbolic geometry, the **angle of parallelism **, is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length *a* between the right angle and the vertex of the angle of parallelism.

In mathematics, **Hilbert's fourth problem** in the 1900 Hilbert problems is a foundational question in geometry. In one statement derived from the original, it was to find — up to an isomorphism — all geometries that have an axiomatic system of the classical geometry, with those axioms of congruence that involve the concept of the angle dropped, and `triangle inequality', regarded as an axiom, added.

**Three-dimensional space** is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the term dimension.

A **Saccheri quadrilateral** is a quadrilateral with two equal sides perpendicular to the base. It is named after Giovanni Gerolamo Saccheri, who used it extensively in his book *Euclides ab omni naevo vindicatus* first published in 1733, an attempt to prove the parallel postulate using the method Reductio ad absurdum.

In geometry, the **Beltrami–Klein model**, also called the **projective model**, **Klein disk model**, and the **Cayley–Klein model**, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere.

In mathematics, the **differential geometry of surfaces** deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: *extrinsically*, relating to their embedding in Euclidean space and *intrinsically*, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

In mathematics, a **conic section** is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though historically it was sometimes called a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.

In geometry, the **Poincaré disk model**, also called the **conformal disk model**, is a model of 2-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk, and the straight lines consist of all circular arcs contained within that disk that are orthogonal to the boundary of the disk, plus all diameters of the disk.

In geometry, **Playfair's axiom** is an axiom that can be used instead of the fifth postulate of Euclid :

In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.

- Heath, Thomas L. (1956),
*The Thirteen Books of Euclid's Elements*(2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.), New York: Dover Publications

- (3 vols.): ISBN 0-486-60088-2 (vol. 1), ISBN 0-486-60089-0 (vol. 2), ISBN 0-486-60090-4 (vol. 3). Heath's authoritative translation plus extensive historical research and detailed commentary throughout the text.

- Richards, Joan L. (1988),
*Mathematical Visions: The Pursuit of Geometry in Victorian England*, Boston: Academic Press, ISBN 0-12-587445-6 - Wilson, James Maurice (1868),
*Elementary Geometry*(1st ed.), London: Macmillan and Co. - Wylie Jr., C. R. (1964),
*Foundations of Geometry*, McGraw–Hill

- Papadopoulos, Athanase; Théret, Guillaume (2014),
*La théorie des parallèles de Johann Heinrich Lambert : Présentation, traduction et commentaires*, Paris: Collection Sciences dans l'histoire, Librairie Albert Blanchard, ISBN 978-2-85367-266-5

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