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单词 LineSegment
释义

line segment


DefinitionSuppose V is a vector space over or , and L is a subset of V.Then L is a line segmentMathworldPlanetmath if L can be parametrizedas

L={a+tbt[0,1]}

for some a,b in V with b0.

Sometimes one needs to distinguish between open and closed (http://planetmath.org/Closed)line segments. Then one defines a closed line segmentas above,and an open line segment as a subset L that can beparametrized as

L={a+tbt(0,1)}

for some a,b in V with b0.

If x and y are two vectors in V and xy, then we denote by[x,y] the set connecting x and y. This is , {αx+(1-α)y|0α1}. One can easily check that [x,y] is a closed line segment.

Remarks

  • An alternative, equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath, definition is as follows:A (closed) line segment is a convex hull of two distinct points.

  • A line segment is connected, non-empty set.

  • If V is a topological vector spaceMathworldPlanetmath, then a closed line segmentis a closed set in V. However, an open line segment isan open set in V if and only if V is one-dimensional.

  • More generally than above, the concept of a line segment can bedefined in an ordered geometry.

Titleline segment
Canonical nameLineSegment
Date of creation2013-03-22 14:19:01
Last modified on2013-03-22 14:19:01
Ownermatte (1858)
Last modified bymatte (1858)
Numerical id12
Authormatte (1858)
Entry typeDefinition
Classificationmsc 03-00
Classificationmsc 51-00
Related topicInterval
Related topicLinearManifold
Related topicLineInThePlane
Related topicCircularSegment
Definesopen line segment
Definesclosed line segment
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更新时间:2025/5/4 18:15:56