line segment

Definition Suppose V is a vector space over or , and L is a subset of V. Then L is a line segmentMathworldPlanetmath if L can be parametrized as


for some a,b in V with b0.

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


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.


  • 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 segment is a closed set in V. However, an open line segment is an 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.

Title line segment
Canonical name LineSegment
Date of creation 2013-03-22 14:19:01
Last modified on 2013-03-22 14:19:01
Owner matte (1858)
Last modified by matte (1858)
Numerical id 12
Author matte (1858)
Entry type Definition
Classification msc 03-00
Classification msc 51-00
Related topic Interval
Related topic LinearManifold
Related topic LineInThePlane
Related topic CircularSegment
Defines open line segment
Defines closed line segment