line segment
Definition
Suppose is a vector space over or , and is a subset of .
Then is a line segment
if can be parametrized
as
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 segment as above, and an open line segment as a subset that can be parametrized 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
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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 |