141.225.85.227 at 22:46, 2 May 2013

2013-05-02T22:46:27Z

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[[Image:Segment definition.svg|thumb|250px|right|The geometric definition of a line segment]]
[[File:Fotothek_df_tg_0003359_Geometrie_%5E_Konstruktion_%5E_Strecke_%5E_Messinstrument.jpg|thumb|historical image – create a line segment (1699)]]
In [[geometry]], a '''line segment''' is a part of a [[line (mathematics)|line]] that is bounded by two distinct end [[Point (geometry)|points]], and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a [[polygon]], the line segment is either an [[edge (geometry)|edge]] (of that polygon) if they are adjacent vertices, or otherwise 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 ''V'' is a [[vector space]] over <math>\mathbb{R}</math> or <math>\mathbb{C}</math>, and ''L'' is a [[subset]] of ''V'', then ''L'' is a '''line segment''' if ''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>, in which case the vectors '''u''' and {{nowrap|'''u''' + '''v'''}} are called the end points of ''L''.

Sometimes one needs to distinguish between "open" and "closed" line segments. Then one defines a '''closed line segment''' as above, and an '''open line segment''' as a subset ''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, it is sometimes defined that a point ''B'' is between two other points ''A'' and ''C'', if the distance ''AB'' added to the distance ''BC'' is equal to the distance ''AC''. Thus in <math>\mathbb{R}^2</math> the line segment with endpoints {{nowrap|''A'' {{=}} (''ax'', ''ay'')}} and {{nowrap|''C'' {{=}} (''cx'', ''cy'')}} is the following collection of points:
:<math>\{ (x,y) | \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}\}</math>.

==Properties==
*A line segment is a [[connected set|connected]], [[non-empty]] [[Set (mathematics)|set]].
*If ''V'' is a [[topological vector space]], then a closed line segment is a [[closed set]] in ''V''. However, an open line segment is an [[open subset|open set]] in ''V'' [[if and only if]] ''V'' is one-dimensional.
*More generally than above, the concept of a line segment can be defined in an [[ordered geometry]].

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

Segments play an important role in other theories. For example, a set is convex if 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. [[Segment addition postulate|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]] in which the semiminor axis goes to zero, the [[Focus (geometry)|foci]] go to the endpoints, and the eccentricity goes to one. As a degenerate orbit this is a [[Elliptic orbit#Radial elliptic trajectory|radial elliptic trajectory]].

==See also==
*[[Interval (mathematics)]]
*[[Line (geometry)]]
*[[Line segment intersection]], the algorithmic problem of finding intersecting pairs in a collection of line segments
*[[Spirangle]]
*[[Segment addition postulate]]
==References==
*David Hilbert: ''The Foundations of Geometry''. The Open Court Publishing Company 1950, p. 4

==External links==
{{commons|Line segment|Line segment}}
{{Wiktionary|line segment}}
*[http://planetmath.org/encyclopedia/LineSegment.html Line Segment at [[PlanetMath]]]
*[http://www.mathopenref.com/linesegment.html Definition of line segment] With interactive animation
*[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}}

[[Category:Elementary geometry]]
[[Category:Linear algebra]]

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141.225.85.227 at 22:46, 2 May 2013