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}}

{{Distinguish|arc (geometry)}}

[[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 – 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 [[Point (geometry)|points]], and contains every point on the line that is between its endpoints. 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 a line above the symbols for the two endpoints (such as <math>\overline{{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>. In which case, the vectors {{math|'''u'''}} and {{nowrapmath|'''u''' + '''v'''}} are called the end points of ''{{mvar|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.