View source for Line segment

{{Short description|Part of a line that is bounded by two distinct end points; line with two endpoints}}
[[Image:Segment definition.svg|thumb|250px|right|The geometric definition of a closed line segment: the [[intersection (Euclidean geometry)|intersection]] of all points at or to the right of {{mvar|A}} with all points at or to the left of {{mvar|B}}]]
[[File:Fotothek df tg 0003359 Geometrie ^ Konstruktion ^ Strecke ^ Messinstrument.jpg|thumb|Historical image of 1699 - creating a line segment]]
{{General geometry}}

In [[geometry]], a '''line segment''' is a part of a [[line (mathematics)|straight line]] that is bounded by two distinct '''endpoints''' (its [[extreme point]]s), and contains every [[Point (geometry)|point]] on the line that is between its endpoints. It is a special case of an ''[[arc (geometry)|arc]]'', with zero [[curvature]]. The [[length]] of a line segment is given by the [[Euclidean distance]] between its endpoints. A '''closed line segment''' includes both endpoints, while an '''open line segment''' excludes both endpoints; a '''half-open line segment''' includes exactly one of the endpoints. In [[geometry]], a line segment is often denoted using an [[overline]] ([[vinculum (symbol)|vinculum]]) above the symbols for the two endpoints, such as in {{mvar|{{overline|AB}}}}.<ref>{{Cite web|title=Line Segment Definition - Math Open Reference|url=https://www.mathopenref.com/linesegment.html|access-date=2020-09-01|website=www.mathopenref.com}}</ref>

Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a [[polygon]] or [[polyhedron]], the line segment is either an [[edge (geometry)|edge]] (of that polygon or polyhedron) if they are adjacent vertices, or a [[diagonal]]. When the end points both lie on a [[curve]] (such as a [[circle]]), a line segment is called a [[chord (geometry)|chord]] (of that curve).

==In real or complex vector spaces==
If {{mvar|V}} is a [[vector space]] over {{tmath|\R}} or {{tmath|\C,}} and {{mvar|L}} is a [[subset]] of {{mvar|V}}, then {{mvar|L}} is a '''line segment''' if {{mvar|L}} can be parameterized as
:<math>L = \{ \mathbf{u} + t\mathbf{v} \mid t \in [0,1]\}</math>

for some vectors <math>\mathbf{u}, \mathbf{v} \in V</math> where {{math|'''v'''}} is nonzero.  The endpoints of {{mvar|L}} are then the vectors {{math|'''u'''}} and {{math|'''u''' + '''v'''}}.

Sometimes, one needs to distinguish between "open" and "closed" line segments. In this case, one would define a '''closed line segment''' as above, and an '''open line segment''' as a subset {{mvar|L}} that can be parametrized as
:<math> L = \{ \mathbf{u}+t\mathbf{v} \mid t\in(0,1)\}</math>

for some vectors <math>\mathbf{u}, \mathbf{v} \in V.</math>

Equivalently, a line segment is the [[convex hull]] of two points. Thus, the line segment can be expressed as a [[convex combination]] of the segment's two end points.

In [[geometry]], one might define point {{mvar|B}} to be between two other points {{mvar|A}} and {{mvar|C}}, if the distance {{mvar|{{abs|AB}}}} added to the distance {{mvar|{{abs|BC}}}} is equal to the distance {{mvar|{{abs|AC}}}}. Thus in {{tmath|\R^2,}} the line segment with endpoints <math>A=(a_x,a_y)</math> and <math>C=(c_x,c_y)</math> is the following collection of points: 
:<math>\Biggl\{ (x,y) \mid \sqrt{(x-c_x)^2 + (y-c_y)^2} + \sqrt{(x-a_x)^2 + (y-a_y)^2} = \sqrt{(c_x-a_x)^2 + (c_y-a_y)^2} \Biggr\} .</math>

==Properties==
*A line segment is a [[connected set|connected]], [[non-empty]] [[Set (mathematics)|set]].
*If {{mvar|V}} is a [[topological vector space]], then a closed line segment is a [[closed set]] in {{mvar|V}}. However, an open line segment is an [[open subset|open set]] in {{mvar|V}} [[if and only if]] {{mvar|V}} is [[One-dimensional space|one-dimensional]].
*More generally than above, the concept of a line segment can be defined in an [[ordered geometry]].
*A pair of line segments can be any one of the following: [[intersection (geometry)|intersecting]], [[parallel (geometry)|parallel]], [[skew lines|skew]], or none of these. The last possibility is a way that line segments differ from lines: if two nonparallel lines are in the same Euclidean plane then they must cross each other, but that need not be true of segments.

==In proofs==
In an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or defined in terms of an [[isometry]] of a line (used as a coordinate system).

Segments play an important role in other theories. For example, in a ''[[convex set]]'', the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of convex sets, to the analysis of a line segment. The ''[[segment addition postulate]]'' can be used to add congruent segment or segments with equal lengths, and consequently substitute other segments into another statement to make segments congruent.

==As a degenerate ellipse==
A line segment can be viewed as a [[Degenerate conic|degenerate case]] of an [[Ellipse#Line segment as a type of degenerate ellipse|ellipse]], in which the semiminor axis goes to zero, the [[Focus (geometry)|foci]] go to the endpoints, and the eccentricity goes to one. A standard definition of an ellipse is the set of points for which the sum of a point's distances to two foci is a constant; if this constant equals the distance between the foci, the line segment is the result. A complete orbit of this ellipse traverses the line segment twice. As a degenerate orbit, this is a [[Elliptic orbit#Radial elliptic trajectory|radial elliptic trajectory]].

==In other geometric shapes==

In addition to appearing as the edges and [[diagonal]]s of [[polygon]]s and [[polyhedron|polyhedra]], line segments also appear in numerous other locations relative to other [[geometric shape]]s.

===Triangles===

Some very frequently considered segments in a [[triangle]] to include the three [[Altitude (geometry)|altitudes]] (each [[perpendicular]]ly connecting a side or its [[extended side|extension]] to the opposite [[vertex (geometry)|vertex]]), the three [[median (geometry)|median]]s (each connecting a side's [[midpoint]] to the opposite vertex), the [[perpendicular bisector]]s of the sides (perpendicularly connecting the midpoint of a side to one of the other sides), and the [[angle bisector|internal angle bisector]]s (each connecting a vertex to the opposite side). In each case, there are various [[equality (mathematics)|equalities]] relating these segment lengths to others (discussed in the articles on the various types of segment), as well as [[list of triangle inequalities|various inequalities]].

Other segments of interest in a triangle include those connecting various [[triangle center]]s to each other, most notably the [[incenter]], the [[circumcenter]], the [[nine-point center]], the [[centroid]] and the [[orthocenter]].

===Quadrilaterals===

In addition to the sides and diagonals of a [[quadrilateral]], some important segments are the two [[quadrilateral#Special line segments|bimedians]] (connecting the midpoints of opposite sides) and the four [[quadrilateral#Special line segments|maltitudes]] (each perpendicularly connecting one side to the midpoint of the opposite side).

===Circles and ellipses===

Any straight line segment connecting  two points on a [[circle]] or [[ellipse]] is called a [[chord (geometry)|chord]]. Any  chord in a circle which has no longer chord is called a [[diameter]], and any segment connecting the circle's [[center (geometry)|center]] (the midpoint of a diameter) to a point on the circle is called a [[radius]].

In an ellipse, the longest chord, which is also the longest [[Diameter#Ellipse|diameter]], is called the ''major axis'', and a segment from the midpoint of the major axis (the ellipse's center) to either endpoint of the major axis is called a ''semi-major axis''.  Similarly, the shortest diameter of an ellipse is called the ''minor axis'', and the segment from its midpoint (the ellipse's center) to either of its endpoints is called a ''semi-minor axis''.  The chords of an ellipse which are [[perpendicular]] to the major axis and pass through one of its [[focus (geometry)|foci]] are called the [[latus rectum|latera recta]] of the ellipse. The ''interfocal segment'' connects the two foci.

==Directed line segment==
{{further|Orientation (vector space)#On a line}}
{{see also|Relative position}}

When a line segment is given an [[orientation (vector space)|orientation]] ([[direction (geometry)|direction]]) it is called a '''directed line segment''' or '''oriented line segment'''. It suggests a [[translation (geometry)|translation]] or [[displacement (geometry)|displacement]] (perhaps caused by a [[force]]). The magnitude and direction are indicative of a potential change. Extending a directed line segment semi-infinitely produces a ''[[directed half-line]]'' and infinitely in both directions produces a ''[[directed line]]''. This suggestion has been absorbed into [[mathematical physics]] through the concept of a [[Euclidean vector]].<ref>Harry F. Davis & Arthur David Snider (1988) ''Introduction to Vector Analysis'', 5th edition, page 1, Wm. C. Brown Publishers {{isbn|0-697-06814-5}}</ref><ref>Matiur Rahman & Isaac Mulolani (2001) ''Applied Vector Analysis'', pages 9 & 10, [[CRC Press]] {{isbn|0-8493-1088-1}}</ref> The collection of all directed line segments is usually reduced by making [[equipollent (geometry)|equipollent]] any pair having the same length and orientation.<ref>Eutiquio C. Young (1978) ''Vector and Tensor Analysis'', pages 2 & 3, [[Marcel Dekker]] {{isbn|0-8247-6671-7}}</ref> This application of an [[equivalence relation]] was introduced by [[Giusto Bellavitis]] in 1835.

==Generalizations==
Analogous to [[straight line]] segments above, one can also define [[Arc (geometry)|arcs]] as segments of a [[curve]].

In one-dimensional space, a ''[[ball (mathematics)|ball]]'' is a line segment.

An [[oriented plane segment]] or ''[[bivector]]'' generalizes the directed line segment.

Beyond Euclidean geometry, [[geodesic segment]]s play the role of line segments.

A line segment is a one-dimensional ''[[simplex]]''; a two-dimensional simplex is a triangle.

==Types of line segments==
* [[Chord (geometry)]]
* [[Diameter]]
* [[Radius]]

==See also==
*[[Polygonal chain]]
*[[Interval (mathematics)]]
*[[Line segment intersection]], the algorithmic problem of finding intersecting pairs in a collection of line segments

==Notes==
{{Reflist}}

==References==
*[[David Hilbert]] ''The Foundations of Geometry''. The Open Court Publishing Company 1950, p.&nbsp;4

==External links==
{{commons|Line segment|Line segment}}
{{Wiktionary|line segment}}
*{{mathworld |urlname=LineSegment |title=Line segment }}
*[http://planetmath.org/linesegment Line Segment] at [[PlanetMath]]
*[http://www.mathopenref.com/constcopysegment.html Copying a line segment with compass and straightedge] 
*[http://www.mathopenref.com/constdividesegment.html Dividing a line segment into N equal parts with compass and straightedge] Animated demonstration

{{PlanetMath attribution|id=5783|title=Line segment}}
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[[Category:Elementary geometry]]
[[Category:Linear algebra]]
[[Category:Line (geometry)]]