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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, parallel lines are coplanar straight lines that do not intersect at any point. Parallel planes are
planes Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
in the same
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
that never meet. ''
Parallel curve A parallel of a curve is the envelope of a family of congruent circles centered on the curve. It generalises the concept of '' parallel (straight) lines''. It can also be defined as a curve whose points are at a constant ''normal distance'' f ...
s'' 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''. Parallel lines are the subject of
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
'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.


Symbol

The parallel symbol is \parallel. For example, AB \parallel CD indicates that line ''AB'' is parallel to line ''CD''. In the
Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, ...
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". The same symbol is used for a binary function in electrical engineering (the
parallel operator The parallel operator (also known as reduced sum, parallel sum or parallel addition) \, (pronounced "parallel", following the parallel lines notation from geometry) is a mathematical function which is used as a shorthand in electrical e ...
). It is distinct from the double-vertical-line brackets that indicate a norm (e.g. as well as from the logical or operator (, , ) in several programming languages.


Euclidean parallelism


Two lines in a plane


Conditions for parallelism

Given parallel straight lines ''l'' and ''m'' in
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, the following properties are equivalent: #Every point on line ''m'' is located at exactly the same (minimum) distance from line ''l'' (''
equidistant A point is said to be equidistant from a set of objects if the distances between that point and each object in the set are equal. In two-dimensional Euclidean geometry, the locus of points equidistant from two given (different) points is th ...
lines''). #Line ''m'' is in the same plane as line ''l'' but does not intersect ''l'' (recall that lines extend to
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions am ...
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. 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 In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
(slope).


History

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 The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postu ...
. Alternative definitions were discussed by other Greeks, often as part of an attempt to prove the parallel postulate.
Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophe ...
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. 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. These reform texts were not without their critics and one of them, Charles Dodgson (a.k.a.
Lewis Carroll Charles Lutwidge Dodgson (; 27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English author, poet and mathematician. His most notable works are '' Alice's Adventures in Wonderland'' (1865) and its sequ ...
), 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 the idea may be traced back to
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
. 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." 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''." 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. 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. 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.


Construction

The three properties above lead to three different methods of construction of parallel lines. image:Par-equi.png, Property 1: Line ''m'' has everywhere the same distance to line ''l''. image:Par-para.png, Property 2: Take a random line through ''a'' that intersects ''l'' in ''x''. Move point ''x'' to infinity. image:Par-perp.png, Property 3: Both ''l'' and ''m'' share a transversal line through ''a'' that intersect them at 90°.


Distance between two parallel lines

Because parallel lines in a Euclidean plane are
equidistant A point is said to be equidistant from a set of objects if the distances between that point and each object in the set are equal. In two-dimensional Euclidean geometry, the locus of points equidistant from two given (different) points is th ...
there is a unique distance between the two parallel lines. Given the equations of two non-vertical, non-horizontal parallel lines, :y = mx+b_1\, :y = mx+b_2\,, 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 :\begin y = mx+b_1 \\ y = -x/m \end and :\begin y = mx+b_2 \\ y = -x/m \end to get the coordinates of the points. The solutions to the linear systems are the points :\left( x_1,y_1 \right)\ = \left( \frac,\frac \right)\, and :\left( x_2,y_2 \right)\ = \left( \frac,\frac \right). 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 :d = \sqrt = \sqrt\,, which reduces to :d = \frac\,. When the lines are given by the general form of the equation of a line (horizontal and vertical lines are included): :ax+by+c_1=0\, :ax+by+c_2=0,\, their distance can be expressed as :d = \frac.


Two lines in three-dimensional space

Two lines in the same
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
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 and a plane

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''.


Two planes

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.


Extension to non-Euclidean geometry

In non-Euclidean geometry, it is more common to talk about
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connecti ...
s 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) 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.


Hyperbolic geometry

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 In hyperbolic geometry, an ideal point, omega point or point at infinity is a well-defined point outside the hyperbolic plane or space. Given a line ''l'' and a point ''P'' not on ''l'', right- and left- limiting parallels to ''l'' through '' ...
), 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 In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line l through a point P not on line R; however, in the plane, two parallels may be closer to l than all others (one in each direction of R) ...
''. As in the illustration through a point ''a'' not on line ''l'' there are two
limiting parallel In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line l through a point P not on line R; however, in the plane, two parallels may be closer to l than all others (one in each direction of R) ...
lines, one for each direction
ideal point In hyperbolic geometry, an ideal point, omega point or point at infinity is a well-defined point outside the hyperbolic plane or space. Given a line ''l'' and a point ''P'' not on ''l'', right- and left- limiting parallels to ''l'' through '' ...
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 In hyperbolic geometry, two lines may intersect, be ultraparallel, or be limiting parallel. The ultraparallel theorem states that every pair of (distinct) ultraparallel lines has a unique common perpendicular (a hyperbolic line which is perpend ...
), and diverge on both sides of this common perpendicular.


Spherical or elliptic geometry

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 In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north ...
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.


Reflexive variant

If ''l, m, n'' are three distinct lines, then l \parallel m \ \land \ m \parallel n \ \implies \ l \parallel n . In this case, parallelism is a
transitive relation In mathematics, a relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Each partial order as well as each equivalence relation needs to be transitive. Definition A hom ...
. However, in case ''l'' = ''n'', the superimposed lines are ''not'' considered parallel in Euclidean geometry. The
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...
between parallel lines is evidently a
symmetric relation A symmetric relation is a type of binary relation. An example is the relation "is equal to", because if ''a'' = ''b'' is true then ''b'' = ''a'' is also true. Formally, a binary relation ''R'' over a set ''X'' is symmetric if: :\forall a, b \in X ...
. According to Euclid's tenets, parallelism is ''not'' a reflexive relation and thus ''fails'' to be an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
. Nevertheless, in affine geometry a
pencil A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage, and keeps it from marking the user's hand. Pencils create marks by physical abrasion, leaving a tra ...
of parallel lines is taken as an
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
in the set of lines where parallelism is an equivalence relation. To this end,
Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing l ...
(1957) adopted a definition of parallelism where two lines are parallel if they have all or none of their points in common.
Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing l ...
(1957
''Geometric Algebra'', page 52
via
Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
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 In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An ''inciden ...
, this variant of parallelism is used in the affine plane.


See also

* Clifford parallel * Collinearity *
Concurrent lines In geometry, lines in a plane or higher-dimensional space are said to be concurrent if they intersect at a single point. They are in contrast to parallel lines. Examples Triangles In a triangle, four basic types of sets of concurrent lines ar ...
*
Limiting parallel In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line l through a point P not on line R; however, in the plane, two parallels may be closer to l than all others (one in each direction of R) ...
*
Parallel curve A parallel of a curve is the envelope of a family of congruent circles centered on the curve. It generalises the concept of '' parallel (straight) lines''. It can also be defined as a curve whose points are at a constant ''normal distance'' f ...
*
Ultraparallel theorem In hyperbolic geometry, two lines may intersect, be ultraparallel, or be limiting parallel. The ultraparallel theorem states that every pair of (distinct) ultraparallel lines has a unique common perpendicular (a hyperbolic line which is perpend ...


Notes


References

* : (3 vols.): (vol. 1), (vol. 2), (vol. 3). Heath's authoritative translation plus extensive historical research and detailed commentary throughout the text. * * *


Further reading

* {{citation , last1 = Papadopoulos , first1 = Athanase , last2=Théret , first2= Guillaume , title = La théorie des parallèles de Johann Heinrich Lambert : Présentation, traduction et commentaires , date = 2014 , publisher = Collection Sciences dans l'histoire, Librairie Albert Blanchard , place=Paris , isbn=978-2-85367-266-5


External links

# Elementary geometry Affine geometry Orientation (geometry)