Line segment: Difference between revisions

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{{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''}}]]

{{Distinguish|arc (geometry)}}

[[File:Fotothek df tg 0003359 Geometrie ^ Konstruktion ^ Strecke ^ Messinstrument.jpg|thumb|historicalHistorical image –of create1699 - creating a line segment (1699)]]

[[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 ''A'' with all points at or to the left of ''B'']]

[[File:Fotothek df tg 0003359 Geometrie ^ Konstruktion ^ Strecke ^ Messinstrument.jpg|thumb|historical image – create a line segment (1699)]]

{{General geometry}}

In [[geometry]], a '''line segment''' is a part of a [[line (mathematics)|straight line]] that is bounded by two distinct end'''endpoints''' (its [[Pointextreme (geometry)|pointspoint]]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 endpointendpoints, 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 aan line[[overline]] ([[vinculum (symbol)|vinculum]]) above the symbols for the two endpoints, (such as <math>\overlinein {{mvar|{{overline|AB}</math>)}}}.<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 <math>\mathbb{{tmath|\R}</math>} or <math>\mathbb{{tmath|\C}</math>,}} 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. In whichThe endpoints of {{mvar|L}} are case,then the vectors {{math|'''u'''}} and {{nowrapmath|'''u''' + '''v'''}} are called the end points of ''L''.

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 <math>{{tmath|\R^2</math>,}} the line segment with endpoints {{nowrap|1=''<math>A'' = (''a_x''a_x, ''a<sub>ya_y)</submath>'')}} and {{nowrap|1=''<math>C'' = (''c_x''c_x, ''c<sub>yc_y)</submath>'')}} is the following collection of points:

:<math>\leftBiggl\{ (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} \rightBiggr\} .</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.

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==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 [[focus (geometry)|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==

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{{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 ''[[raydirected (geometry)|rayhalf-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 "equivalent"[[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]] dateswas fromintroduced by [[Giusto Bellavitis]]'s introduction of the concept of [[equipollence (geometry)|equipollence]] of directed line segments in 1835.

==Generalizations==

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

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

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==

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==External links==

{{commons|Line segment|Line segment}}

{{Wiktionary|line segment}}

*{{mathworld |urlname=LineSegment |title=Line segment }}

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[[Category:Elementary geometry]]

[[Category:Linear algebra]]

[[Category:Line (geometry)]]