Editing Line (geometry) (section)

== Properties ==
In the [[Greek mathematics|Greek]] deductive geometry of [[Euclid's Elements|Euclid's ''Elements'']], a general ''line'' (now called a ''[[curve]]'') is defined as a "breadthless length", and a ''straight line'' (now called a [[line segment]]) was defined as a line "which lies evenly with the points on itself".<ref name=":0">{{citation |last=Faber |first=Richard L. |title=Foundations of Euclidean and Non-Euclidean Geometry |year=1983 |location=New York |publisher=Marcel Dekker |isbn=0-8247-1748-1}}</ref>{{Rp|page=291}} These definitions appeal to readers' physical experience, relying on terms that are not themselves defined, and the definitions are never explicitly referenced in the remainder of the text. In modern geometry, a line is usually either taken as a [[primitive notion]] with properties given by [[axiom]]s,<ref name=":0" />{{Rp|page=95}} or else defined as a [[set (mathematics)|set]] of points obeying a linear relationship, for instance when [[real number]]s are taken to be primitive and geometry is established [[analytic geometry|analytically]] in terms of numerical [[Cartesian coordinate system|coordinates]].

In an axiomatic formulation of Euclidean geometry, such as that of [[Hilbert's axioms|Hilbert]] (modern mathematicians added to Euclid's original axioms to fill perceived logical gaps),<ref name=":0" />{{Rp|page=108}} a line is stated to have certain properties that relate it to other lines and [[point (geometry)|points]]. For example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect at most at one point.<ref name=":0" />{{Rp|page=300}} In two [[dimension]]s (i.e., the Euclidean [[plane (mathematics)|plane]]), two lines that do not intersect are called [[Parallel (geometry)|parallel]]. In higher dimensions, two lines that do not intersect are parallel if they are contained in a [[Plane (geometry)|plane]], or [[Skew lines|skew]] if they are not.

On a [[Euclidean plane]], a line can be represented as a boundary between two regions.<ref>{{Cite book |last=Foster |first=Colin |url=https://www.worldcat.org/oclc/747274805 |title=Resources for teaching mathematics, 14–16 |date=2010 |publisher=Continuum International Pub. Group |isbn=978-1-4411-3724-1 |location=New York |oclc=747274805}}</ref>{{Rp|page=104}} Any collection of finitely many lines partitions the plane into [[convex polygon]]s (possibly unbounded); this partition is known as an [[arrangement of lines]].

===In higher dimensions===
In [[three-dimensional space]], a [[first degree equation]] in the variables ''x'', ''y'', and ''z'' defines a plane, so two such equations, provided the planes they give rise to are not parallel, define a line which is the intersection of the planes. More generally, in ''n''-dimensional space ''n''−1 first-degree equations in the ''n'' [[Cartesian coordinate system|coordinate]] variables define a line under suitable conditions.

In more general [[Euclidean space]], '''R'''<sup>''n''</sup> (and analogously in every other [[affine space]]), the line ''L'' passing through two different points ''a'' and ''b'' is the subset
<math display="block">L = \left\{ (1 - t) \, a + t b \mid t\in\mathbb{R}\right\}.</math>
The [[direction (geometry)|direction]] of the line is from a reference point ''a'' (''t'' = 0) to another point ''b'' (''t'' = 1), or in other words, in the direction of the vector ''b''&nbsp;−&nbsp;''a''. Different choices of ''a'' and ''b'' can yield the same line.

====Collinear points====
{{Main|Collinearity}}

Three or more points are said to be ''collinear'' if they lie on the same line. If three points are not collinear, there is exactly one [[plane (geometry)|plane]] that contains them.

In [[affine coordinates]], in ''n''-dimensional space the points ''X'' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>), ''Y'' = (''y''<sub>1</sub>, ''y''<sub>2</sub>, ..., ''y''<sub>''n''</sub>), and ''Z'' = (''z''<sub>1</sub>, ''z''<sub>2</sub>, ..., ''z''<sub>''n''</sub>) are collinear if the [[matrix (mathematics)|matrix]]
<math display="block">\begin{bmatrix}
 1 & x_1 & x_2 & \cdots & x_n \\
 1 & y_1 & y_2 & \cdots & y_n \\
 1 & z_1 & z_2 & \cdots & z_n
\end{bmatrix}</math>
has a [[rank (linear algebra)|rank]] less than 3. In particular, for three points in the plane (''n'' = 2), the above matrix is square and the points are collinear if and only if its [[determinant]] is zero.

Equivalently for three points in a plane, the points are collinear if and only if the slope between one pair of points equals the slope between any other pair of points (in which case the slope between the remaining pair of points will equal the other slopes). By extension, ''k'' points in a plane are collinear if and only if any (''k''–1) pairs of points have the same pairwise slopes.

In [[Euclidean geometry]], the [[Euclidean distance]] ''d''(''a'',''b'') between two points ''a'' and ''b'' may be used to express the collinearity between three points by:<ref>{{cite book |author-link=Alessandro Padoa |last=Padoa |first=Alessandro |title=Un nouveau système de définitions pour la géométrie euclidienne |language=fr |publisher=[[International Congress of Mathematicians]] |date=1900}}</ref><ref>{{cite book |author-link=Bertrand Russell |last=Russell |first=Bertrand |title=[[The Principles of Mathematics]] |page=410}}</ref>
:The points ''a'', ''b'' and ''c'' are collinear if and only if ''d''(''x'',''a'') = ''d''(''c'',''a'') and ''d''(''x'',''b'') = ''d''(''c'',''b'') implies ''x'' = ''c''.
However, there are other notions of distance (such as the [[Manhattan distance]]) for which this property is not true.

In the geometries where the concept of a line is a [[primitive notion]], as may be the case in some [[synthetic geometry|synthetic geometries]], other methods of determining collinearity are needed.

=== Relationship with other figures===
[[File:Tangent to a curve.svg|alt=see caption|thumb|Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot.]]
In Euclidean geometry, all lines are [[congruence (geometry)|congruent]], meaning that every line can be obtained by moving a specific line. However, lines may play special roles with respect to other [[geometric object]]s and can be classified according to that relationship. 

For instance, with respect to a [[Conic section|conic]] (a [[circle]], [[ellipse]], [[parabola]], or [[hyperbola]]), lines can be:
* [[tangent line]]s, which touch the conic at a single point;
* [[secant line]]s, which intersect the conic at two points and pass through its interior;<ref name="cag">{{citation |last1=Protter |first1=Murray H. |title=Calculus with Analytic Geometry |url=https://books.google.com/books?id=jTmuOwwGDwoC&pg=PA62 |page=62 |year=1988 |publisher=Jones & Bartlett Learning |isbn=9780867200935 |last2=Protter |first2=Philip E. |author1-link=Murray H. Protter}}</ref>
* exterior lines, which do not meet the conic at any point of the Euclidean plane; or
* a [[Directrix of a conic section|directrix]], whose distance from a point helps to establish whether the point is on the conic.
* a [[coordinate line]], a linear coordinate dimension

In the context of determining [[parallel (geometry)|parallelism]] in Euclidean geometry, a [[transversal (geometry)|transversal]] is a line that intersects two other lines that may or not be parallel to each other.

For more general [[algebraic curve]]s, lines could also be:
* ''i''-secant lines, meeting the curve in ''i'' points counted without multiplicity, or
* [[asymptote]]s, which a curve approaches arbitrarily closely without touching it.<ref>{{citation |last=Nunemacher |first=Jeffrey |title=Asymptotes, Cubic Curves, and the Projective Plane |journal=Mathematics Magazine |volume=72 |issue=3 |pages=183–192 |year=1999 |citeseerx=10.1.1.502.72 |doi=10.2307/2690881 |jstor=2690881}}</ref>
With respect to [[Euclidean triangle|triangles]] we have:
* the [[Euler line]],
* the [[Simson line]]s, and
* [[central line (geometry)|central lines]].

For a [[convex polygon|convex]] [[quadrilateral]] with at most two parallel sides, the [[Newton line]] is the line that connects the midpoints of the two [[diagonal]]s.<ref name="Alsina">{{cite book |first1=Claudi |last1=Alsina |first2=Roger B. |last2=Nelsen |title=Charming Proofs: A Journey Into Elegant Mathematics |publisher=MAA |date=2010 |isbn=9780883853481 |pages=108–109}} ({{Google books|mIT5-BN_L0oC|online copy|page=108}})</ref>

For a [[hexagon]] with vertices lying on a conic we have the [[Pascal line]] and, in the special case where the conic is a pair of lines, we have the [[Pappus's hexagon theorem|Pappus line]].

[[Parallel (geometry)|Parallel lines]] are lines in the same plane that never cross. [[Line-line intersection|Intersecting lines]] share a single point in common. Coincidental lines coincide with each other—every point that is on either one of them is also on the other.

[[Perpendicular lines]] are lines that intersect at [[right angle]]s.<ref>{{citation |last1=Kay |first1=David C. |title=College Geometry |page=114 |year=1969 |location=New York |publisher=[[Holt, Rinehart and Winston]] |isbn=978-0030731006 |lccn=69-12075 |oclc=47870}}</ref>

In [[three-dimensional space]], [[skew lines]] are lines that are not in the same plane and thus do not intersect each other.

=== In axiomatic systems ===
In [[synthetic geometry]], the concept of a line is often considered as a [[primitive notion]],<ref name=":0" />{{Rp|page=95}} meaning it is not being defined by using other concepts, but  it is defined by the properties, called  [[axiom]]s, that it must satisfy.<ref>{{citation |last=Coxeter |first=H.S.M |title=Introduction to Geometry |url=https://archive.org/details/introductiontoge0002coxe |page=4 |year=1969 |edition=2nd |place=New York |publisher=John Wiley & Sons |isbn=0-471-18283-4 |url-access=registration}}</ref> 

However, the [[axiomatic]] definition of a line does not explain the relevance of the concept and is often too abstract for beginners. So, the definition is often replaced or completed by a ''mental image'' or ''intuitive description'' that allows understanding what is a line. Such descriptions are sometimes referred to as definitions, but are not true definitions since they cannot used in [[mathematical proof]]s. The "definition" of line in [[Euclid's Elements]] falls into this category;<ref name=":0" />{{Rp|page=95}} and is never used in proofs of theorems.