Line segment
From Wikipedia, the free encyclopedia
In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment is either an edge (of that polygon) if they are adjacent vertices, or otherwise a diagonal. When the end points both lie on a curve such as a circle, a line segment is called a chord (of that curve).
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[edit] Definition
If 
is a vector space over 
or 
, and 
is a subset of 
then is a line segment if can be parameterized as
![L = \{ \mathbf{u}+t\mathbf{v} \mid t\in[0,1]\}](http://upload.wikimedia.org/math/1/1/2/11258fa7f33e01a97f78ba558cf5c862.png)
for some vectors 
, in which case the vectors 
and 
are called the end points of 
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 that can be parametrized as
for some vectors .
An alternative, equivalent, definition is as follows: A (closed) line segment is a convex hull of two points.
[edit] Properties
- A line segment is a connected, non-empty set.
- If V is a topological vector space, 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.
[edit] In Proofs
In geometry, the segment addition postulate states that if B is between A and C, then segment AB + segment BC = segment AC.
[edit] See also
- Interval (mathematics)
- Line (geometry)
- Line segment intersection, the algorithmic problem of finding intersecting pairs in a collection of line segments
[edit] References
- David Hilbert: The Foundations of Geometry. The Open Court Publishing Company 1950, p. 4
[edit] External links
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Wikimedia Commons has media related to: Line segment |
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Look up line segment in Wiktionary, the free dictionary. |
- Line Segment at PlanetMath
- Definition of line segment With interactive animation
- Copying a line segment with compass and straightedge
- Dividing a line segment into N equal parts with compass and straightedge Animated demonstration
This article incorporates material from Line segment on PlanetMath, which is licensed under the GFDL.
