View source for Line (geometry)

{{Short description|Straight figure with zero width and depth}}
[[File:Gerade.svg|alt=see caption|thumb|290x290px|A red line near the [[Origin (mathematics)|origin]] on the two-dimensional [[Cartesian coordinate system|Cartesian coordinate system]]]]
{{General geometry |1d}}

In [[Geometry|geometry]], a '''straight line''', usually abbreviated '''line''', is an infinitely long object with no width, depth, or [[Curvature|curvature]], an idealization of such physical objects as a [[Straightedge|straightedge]], a taut string, or a [[Physics:Ray (optics)|ray of light]]. Lines are [[Space (mathematics)|space]]s of [[Physics:Dimension|dimension]] one, which may be [[Embedding|embedded]] in spaces of dimension two, three, or higher. The word ''line'' may also refer, in everyday life, to a [[Line segment|line segment]], which is a part of a line delimited by two [[Point (geometry)|points]] (its ''endpoints'').

[[Euclid's Elements|Euclid's ''Elements'']] defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several postulates as basic unprovable properties on which the rest of geometry was established. '''''Euclidean line''''' and ''[[Euclidean geometry]]'' are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as [[Non-Euclidean geometry|non-Euclidean]], [[Projective geometry|projective]], and [[Affine geometry|affine geometry]].

== Properties ==
In the Greek deductive geometry of [[Euclid's Elements|Euclid's ''Elements'']], a general ''line'' (now called a ''[[Curve|curve]]'') is defined as a "breadthless length", and a ''straight line'' (now called a [[Line segment|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|primitive notion]] with properties given by [[Axiom|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|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 (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 [[Physics:Dimension|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|convex polygon]]s (possibly unbounded); this partition is known as an [[Arrangement of lines|arrangement of lines]].

===In higher dimensions===
In [[Three-dimensional space|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|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 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 points are said to be ''collinear'' if they lie on the same line. Three points ''[[General position|usually]]'' determine a [[Plane (geometry)|plane]], but in the case of three collinear points this does ''not'' happen.

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|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 |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|primitive notion]], as may be the case in some [[Synthetic geometry|synthetic geometries]], other methods of determining collinearity are needed.

=== Types ===
[[File:Tangent_to_a_curve.svg|alt=see caption|thumb|220x220px|Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot.]]
In a sense,{{efn|Technically, the collineation group acts transitively on the set of lines.}} all lines in Euclidean geometry are equal, in that, without coordinates, one can not tell them apart from one another. However, lines may play special roles with respect to other objects in the geometry and be divided into types according to that relationship. For instance, with respect to a [[Conic section|conic]] (a [[Circle|circle]], [[Ellipse|ellipse]], [[Parabola|parabola]], or [[Hyperbola|hyperbola]]), lines can be:
* tangent lines, which touch the conic at a single point;
* [[Secant line|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. |}}.</ref>
* exterior lines, which do not meet the conic at any point of the Euclidean plane; or
* a directrix, whose distance from a point helps to establish whether the point is on the conic.
* a [[Coordinate line|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|algebraic curve]]s, lines could also be:
* ''i''-secant lines, meeting the curve in ''i'' points counted without multiplicity, or
* [[Asymptote|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 triangles we have:
* the [[Euler line]],
* the [[Simson line]]s, and
* [[Central line (geometry)|central lines]].

For a [[Convex polygon|convex]] [[Quadrilateral|quadrilateral]] with at most two parallel sides, the [[Newton line]] is the line that connects the midpoints of the two [[Diagonal|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|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. 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|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|three-dimensional space]], [[Skew lines|skew lines]] are lines that are not in the same plane and thus do not intersect each other.

=== In axiomatic systems ===
The concept of line is often considered in geometry as a [[Primitive notion|primitive notion]] in [[Axiomatic system|axiomatic systems]],<ref name=":0" />{{Rp|page=95}} meaning it is not being defined by other concepts.<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> In those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the properties of lines are dictated by the [[Axiom|axiom]]s which they must satisfy.

In a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance, it is possible to provide a ''description'' or ''mental image'' of a primitive notion, to give a foundation to build the notion on which would formally be based on the (unstated) axioms. Descriptions of this type may be referred to, by some authors, as definitions in this informal style of presentation. These are not true definitions, and could not be used in formal proofs of statements. The "definition" of line in [[Euclid's Elements]] falls into this category.<ref name=":0" />{{Rp|page=95}} Even in the case where a specific geometry is being considered (for example, [[Euclidean geometry]]), there is no generally accepted agreement among authors as to what an informal description of a line should be when the subject is not being treated formally.

== Definition ==
{{Main|Line coordinates}}

=== Linear equation ===
{{Main|Linear equation}}
[[File:Linear_Function_Graph.svg|alt=y = –x + 5 (going down) and y = 0.5x + 2 (rising up slower)|thumb|Line graphs of linear equations on the Cartesian plane ]]
Lines in a Cartesian plane or, more generally, in affine coordinates, are characterized by linear equations. More precisely, every line <math>L</math> (including vertical lines) is the set of all points whose coordinates (''x'', ''y'') satisfy a linear equation; that is,
<math display="block">L = \{(x,y)\mid ax+by=c\}, </math>
where ''a'', ''b'' and ''c'' are fixed [[Real number|real number]]s (called [[Coefficient|coefficient]]s) such that ''a'' and ''b'' are not both zero. Using this form, vertical lines correspond to equations with ''b'' = 0.

One can further suppose either {{math|1=''c'' = 1}} or {{math|1=''c'' = 0}}, by dividing everything by {{mvar|c}} if it is not zero.

There are many variant ways to write the equation of a line which can all be converted from one to another by algebraic manipulation. The above form is sometimes called the ''standard form''. If the constant term is put on the left, the equation becomes
<math display="block">ax + by - c = 0,</math>
and this is sometimes called the ''general form'' of the equation. However, this terminology is not universally accepted, and many authors do not distinguish these two forms.

These forms are generally named by the type of information (data) about the line that is needed to write down the form. Some of the important data of a line is its slope, [[Root of a function|x-intercept]], known points on the line and y-intercept.

The equation of the line passing through two different points <math>P_0( x_0, y_0 )</math> and <math>P_1(x_1, y_1)</math> may be written as
<math display="block">(y - y_0)(x_1 - x_0) = (y_1 - y_0)(x - x_0).</math>
If {{math|''x''<sub>0</sub> ≠ ''x''<sub>1</sub>}}, this equation may be rewritten as
<math display="block">y=(x-x_0)\,\frac{y_1-y_0}{x_1-x_0}+y_0</math>
or
<math display="block">y=x\,\frac{y_1-y_0}{x_1-x_0}+\frac{x_1y_0-x_0y_1}{x_1-x_0}\,.</math>In [[Plane (mathematics)|two dimensions]], the equation for non-vertical lines is often given in the ''slope–intercept form'':

<math display="block"> y = mx + b </math>
where:
* ''m'' is the [[Slope|slope]] or [[Slope|gradient]] of the line.
* ''b'' is the [[Y-intercept|y-intercept]] of the line.
* ''x'' is the independent variable of the function {{math|1=''y'' = ''f''(''x'')}}.

The slope of the line through points <math>A(x_a, y_a)</math> and <math>B(x_b, y_b)</math>, when <math>x_a \neq x_b</math>, is given by <math>m = (y_b - y_a)/(x_b - x_a)</math> and the equation of this line can be written <math>y = m (x - x_a) + y_a</math>.

As a note, lines in three dimensions may also be described as the simultaneous solutions of two [[Linear equation|linear equation]]s
<math display="block"> a_1 x + b_1 y + c_1 z - d_1 = 0 </math>
<math display="block"> a_2 x + b_2 y + c_2 z - d_2 = 0 </math>
such that <math> (a_1,b_1,c_1)</math> and <math> (a_2,b_2,c_2)</math> are not proportional (the relations <math> a_1 = t a_2, b_1 = t b_2, c_1 = t c_2 </math> imply <math>t = 0</math>). This follows since in three dimensions a single linear equation typically describes a [[Plane (geometry)|plane]] and a line is what is common to two distinct intersecting planes.

=== Parametric equation ===
{{Further|Parametric equation}}
Parametric equations are also used to specify lines, particularly in those in [[Three-dimensional space|three dimensions]] or more because in more than two dimensions lines ''cannot'' be described by a single linear equation.

In three dimensions lines are frequently described by parametric equations:
<math display="block">\begin{align}
x &= x_0 + at \\
y &= y_0 + bt \\
z &= z_0 + ct
\end{align}</math>
where:
* ''x'', ''y'', and ''z'' are all functions of the independent variable ''t'' which ranges over the real numbers.
* (''x''<sub>0</sub>, ''y''<sub>0</sub>, ''z''<sub>0</sub>) is any point on the line.
* ''a'', ''b'', and ''c'' are related to the slope of the line, such that the direction vector (''a'', ''b'', ''c'') is parallel to the line.

Parametric equations for lines in higher dimensions are similar in that they are based on the specification of one point on the line and a direction vector.

=== Hesse normal form ===
{{main|Hesse normal form}}
[[File:Hesse_normalenform.svg|thumb|Distance from the origin O to the line E calculated with the Hesse normal form. Normal vector in red, line in green, point O shown in blue.]]
The ''normal form'' (also called the ''Hesse normal form'',<ref>{{citation|title=Plane Analytic Geometry: With Introductory Chapters on the Differential Calculus|first=Maxime|last=Bôcher|publisher=H. Holt|year=1915| page=44| url=https://books.google.com/books?id=bYkLAAAAYAAJ&pg=PA44|url-status=live|archive-url=https://web.archive.org/web/20160513124511/https://books.google.com/books?id=bYkLAAAAYAAJ&pg=PA44|archive-date=2016-05-13}}.</ref> after the German mathematician [[Biography:Otto Hesse|Ludwig Otto Hesse]]), is based on the ''[[Normal (geometry)|normal]] segment'' for a given line, which is defined to be the line segment drawn from the [[Origin (mathematics)|origin]] perpendicular to the line. This segment joins the origin with the closest point on the line to the origin. The normal form of the equation of a straight line on the plane is given by:
<math display="block"> x \cos \varphi + y \sin \varphi - p = 0 ,</math>
where <math>\varphi</math> is the angle of inclination of the normal segment (the oriented angle from the unit vector of the {{math|''x''}}-axis to this segment), and {{math|''p''}} is the (positive) length of the normal segment. The normal form can be derived from the standard form <math>ax + by = c</math> by dividing all of the coefficients by
<math display="block">\frac{c}{|c|}\sqrt{a^2 + b^2}.</math>

Unlike the slope-intercept and intercept forms, this form can represent any line but also requires only two finite parameters, <math>\varphi</math> and {{math|''p''}}, to be specified. If {{math|''p'' > 0}}, then <math>\varphi</math> is uniquely defined modulo {{math|2''π''}}. On the other hand, if the line is through the origin ({{math|1=''c'' = ''p'' = 0}}), one drops the {{math|''c''/{{abs|''c''}}}} term to compute <math>\sin\varphi</math> and <math>\cos\varphi</math>, and it follows that <math>\varphi</math> is only defined modulo {{pi}}.

== Other representations ==

===Vectors===
The vector equation of the line through points A and B is given by <math>\mathbf{r} = \mathbf{OA} + \lambda\, \mathbf{AB}</math> (where λ is a [[Scalar (mathematics)|scalar]]).

If '''a''' is vector '''OA''' and '''b''' is vector '''OB''', then the equation of the line can be written: <math>\mathbf{r} = \mathbf{a} + \lambda (\mathbf{b} - \mathbf{a})</math>.

A ray starting at point ''A'' is described by limiting λ. One ray is obtained if λ ≥ 0, and the opposite ray comes from λ ≤ 0.

===Polar coordinates===
[[File:Parametres polaires droite.svg|alt=see caption|thumb|A line on polar coordinates without passing though the origin, with the general parametric equation written above]]
In a Cartesian plane, polar coordinates {{math|(''r'', ''θ'')}} are related to Cartesian coordinates by the parametric equations:<ref>{{Cite book |last1=Torrence |first1=Bruce F. |title=The Student's Introduction to MATHEMATICA: A Handbook for Precalculus, Calculus, and Linear Algebra |last2=Torrence |first2=Eve A. |date=29 Jan 2009 |publisher={{wipe|Cambridge University Press}} |isbn=9781139473736 |pages=314}}</ref><math display="block">x=r\cos\theta, \quad y=r\sin\theta.</math>

In polar coordinates, the equation of a line not passing through the [[Origin (mathematics)|origin]]—the point with coordinates {{math|(0, 0)}}—can be written
<math display="block">r = \frac p {\cos (\theta-\varphi)},</math>
with {{math|''r'' > 0}} and <math>\varphi-\pi/2 < \theta < \varphi + \pi/2.</math> Here, {{mvar|p}} is the (positive) length of the [[Line segment|line segment]] perpendicular to the line and delimited by the origin and the line, and <math>\varphi</math> is the (oriented) angle from the {{mvar|x}}-axis to this segment.

It may be useful to express the equation in terms of the angle <math>\alpha=\varphi+\pi/2</math> between the {{mvar|x}}-axis and the line. In this case, the equation becomes
<math display="block">r=\frac p {\sin (\theta-\alpha)},</math>
with {{math|''r'' > 0}} and <math>0 < \theta < \alpha + \pi.</math>

These equations can be derived from the normal form of the line equation by setting <math>x = r \cos\theta,</math> and <math>y = r \sin\theta,</math> and then applying the angle difference identity for sine or cosine.

These equations can also be proven [[Geometry|geometrically]] by applying [[Trigonometric functions#Right-angled triangle definitions|right triangle definitions]] of sine and cosine to the [[Right triangle|right triangle]] that has a point of the line and the origin as vertices, and the line and its perpendicular through the origin as sides.

The previous forms do not apply for a line passing through the origin, but a simpler formula can be written: the polar coordinates <math>(r, \theta)</math> of the points of a line passing through the origin and making an angle of <math>\alpha</math> with the {{mvar|x}}-axis, are the pairs <math>(r, \theta)</math> such that 
<math display="block">r\ge 0,\qquad \text{and} \quad \theta=\alpha \quad\text{or}\quad \theta=\alpha +\pi.</math>

==Generalizations of the Euclidean line {{anchor|Generalizations}}==

In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in [[Analytic geometry|analytic geometry]], a line in the plane is often defined as the set of points whose coordinates satisfy a given [[Linear equation|linear equation]], but in a more abstract setting, such as [[Incidence geometry|incidence geometry]], a line may be an independent object, distinct from the set of points which lie on it.

When a geometry is described by a set of [[Axiom|axiom]]s, the notion of a line is usually left undefined (a so-called [[Primitive notion|primitive]] object). The properties of lines are then determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry. Thus in [[Differential geometry|differential geometry]], a line may be interpreted as a [[Geodesic|geodesic]] (shortest path between points), while in some [[Projective geometry|projective geometries]], a line is a 2-dimensional vector space (all linear combinations of two independent vectors). This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line.

=== Projective geometry ===
{{Further|Geodesic}}
[[File:Great_circle_hemispheres.png|right|thumb|A great circle divides the sphere in two equal hemispheres, while also satisfying the "no curvature" property.]]
In many models of [[Projective geometry|projective geometry]], the representation of a line rarely conforms to the notion of the "straight curve" as it is visualised in Euclidean geometry. In [[Elliptic geometry|elliptic geometry]] we see a typical example of this.<ref name=":0" />{{Rp|page=108}} In the spherical representation of elliptic geometry, lines are represented by [[Great circle|great circle]]s of a sphere with diametrically opposite points identified. In a different model of elliptic geometry, lines are represented by Euclidean [[Plane (geometry)|planes]] passing through the origin. Even though these representations are visually distinct, they satisfy all the properties (such as, two points determining a unique line) that make them suitable representations for lines in this geometry.

The "shortness" and "straightness" of a line, interpreted as the property that the [[Distance|distance]] along the line between any two of its points is minimized (see [[Triangle inequality|triangle inequality]]), can be generalized and leads to the concept of [[Geodesic|geodesic]]s in [[Metric space|metric space]]s.

==Extensions==

=== Ray ===<!-- This section is linked (older copy) from: [[Tangent]], Ray (mathematics),  -->
 [[File:Ray (A, B, C).svg|300x300px|A ray with a terminus at A, with two points B and C on the right|alt=Ray|thumb]]

Given a line and any point ''A'' on it, we may consider ''A'' as decomposing this line into two parts.
Each such part is called a '''ray''' and the point ''A'' is called its ''initial point''. It is also known as half-line, a one-dimensional [[Half-space (geometry)|half-space]]. The point A is considered to be a member of the ray.{{efn|On occasion we may consider a ray without its initial point. Such rays are called ''open'' rays, in contrast to the typical ray which would be said to be ''closed''.}} Intuitively, a ray consists of those points on a line passing through ''A'' and proceeding indefinitely, starting at ''A'', in one direction only along the line. However, in order to use this concept of a ray in proofs a more precise definition is required.

Given distinct points ''A'' and ''B'', they determine a unique ray with initial point ''A''. As two points define a unique line, this ray consists of all the points between ''A'' and ''B'' (including ''A'' and ''B'') and all the points ''C'' on the line through ''A'' and ''B'' such that ''B'' is between ''A'' and ''C''.<ref>{{citation |last=Wylie Jr. |first=C.R. |title=Foundations of Geometry |year=1964 |at=p. 59, definition 3 |place=New York |publisher=McGraw-Hill |isbn=0-07-072191-2}}</ref> This is, at times, also expressed as the set of all points ''C'' on the line determined by ''A'' and ''B'' such that ''A'' is not between ''B'' and ''C''.<ref>{{citation |last=Pedoe |first=Dan |title=Geometry: A Comprehensive Course |page=2 |year=1988 |place=Mineola, NY |publisher=Dover |isbn=0-486-65812-0}}</ref> A point ''D'', on the line determined by ''A'' and ''B'' but not in the ray with initial point ''A'' determined by ''B'', will determine another ray with initial point ''A''. With respect to the ''AB'' ray, the ''AD'' ray is called the ''opposite ray''.

Thus, we would say that two different points, ''A'' and ''B'', define a line and a decomposition of this line into the [[Disjoint union|disjoint union]] of an open segment {{open-open|''A'', ''B''}} and two rays, ''BC'' and ''AD'' (the point ''D'' is not drawn in the diagram, but is to the left of ''A'' on the line ''AB''). These are not opposite rays since they have different initial points.

In Euclidean geometry two rays with a common endpoint form an [[Angle|angle]].<ref>{{SpringerEOM|mode=cs2|last=Sidorov|first=L. A.|date=2001|id=Angle&oldid=13323|title=Angle}}</ref>

The definition of a ray depends upon the notion of betweenness for points on a line. It follows that rays exist only for geometries for which this notion exists, typically [[Euclidean geometry]] or [[Affine geometry|affine geometry]] over an [[Ordered field|ordered field]]. On the other hand, rays do not exist in [[Projective geometry|projective geometry]] nor in a geometry over a non-ordered field, like the [[Complex number|complex number]]s or any [[Finite field|finite field]].

===Line segment===
{{main|Line segment}}
[[File:Adcinak.svg|alt=see caption|thumb|Drawing of a line segment "AB" on the line "a"]]
A [[Line segment|line segment]] is a part of a line that is bounded by two distinct end points and contains every point on the line between its end points. Depending on how the line segment is defined, either of the two end points may or may not be part of the line segment. Two or more line segments may have some of the same relationships as lines, such as being parallel, intersecting, or skew, but unlike lines they may be none of these, if they are coplanar and either do not intersect or are [[Collinearity|collinear]].

=== Number line ===
{{main|Number line}}
[[File:Number line with x smaller than y.svg|alt=see caption|thumb|A number line, with variable x on the left and y on the right. Therefore, x is smaller than y.|300x300px]]
A point on number line corresponds to a [[Real number|real number]] and vice versa.<ref>{{cite book |last1=Stewart |first1=James B. |title=College Algebra |last2=Redlin |first2=Lothar |last3=Watson |first3=Saleem |publisher=Brooks Cole |year=2008 |isbn=978-0-495-56521-5 |edition=5th |pages=13&ndash;19 |authorlink=James Stewart (mathematician)}}</ref> Usually, [[Integer|integers]] are evenly spaced on the line, with positive numbers are on the right, negative numbers on the left. As an extension to the concept, an [[Imaginary line (mathematics)|imaginary line]] representing [[Imaginary number|imaginary numbers]] can be drawn perpendicular to the number line at zero.<ref>{{citation |last=Patterson |first=B. C. |title=The inversive plane |journal=The American Mathematical Monthly |volume=48 |pages=589–599 |year=1941 |issue=9 |doi=10.2307/2303867 |jstor=2303867 |mr=0006034 |authorlink=Boyd Patterson}}.</ref> The two lines forms the [[Complex plane|complex plane]], a geometrical representation of the set of [[Complex number|complex numbers]].

==See also==
* [[Affine transformation]]
* [[Curve]]
* [[Distance between two parallel lines]]
* [[Distance from a point to a line]]
* [[Incidence (geometry)]]
* [[Line segment]]
* [[Generalised circle]]
* [[Locus (mathematics)|Locus]]
* [[Plane (geometry)]]
* Polyline

== Notes ==
{{notelist}}

== References ==
{{Reflist}}

==External links==
* {{springer|title=Line (curve)|id=p/l059020}}
* [http://www.cut-the-knot.org/Curriculum/Calculus/StraightLine.shtml Equations of the Straight Line] at Cut-the-Knot


{{DEFAULTSORT:Line (Geometry)}}
[[Category:Elementary geometry]]
[[Category:Analytic geometry]]
[[Category:Mathematical concepts]]
[[Category:Infinity]]

{{Sourceattribution|Line (geometry)}}