line segment
DefinitionSuppose is a vector space over or , and is a subset of .Then is a line segment if can be parametrizedas
for some in with .
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 that can beparametrized as
for some in with .
If and are two vectors in and , then we denote by the set connecting and . This is , . One can easily check that is a closed line segment.
Remarks
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An alternative, equivalent
, definition is as follows:A (closed) line segment is a convex hull of two distinct points.
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A line segment is connected, non-empty set.
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If is a topological vector space
, then a closed line segmentis a closed set in . However, an open line segment isan open set in if and only if is one-dimensional.
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More generally than above, the concept of a line segment can bedefined 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 |