Line segment

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The geometric definition of a line segment
The geometric definition of a line segment

In geometry, a line segment is a part of a line that is bounded by two distinct 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 V\,\! is a vector space over \mathbb{R} or \mathbb{C}, and L\,\! is a subset of V,\,\! then L\,\! is a line segment if L\,\! can be parametrized as

 L = \{ \mathbf{u}+t\mathbf{v} \mid t\in[0,1]\}

for some vectors \mathbf{u}, \mathbf{v} \in V\,\! with  \mathbf{v} \neq \mathbf{0}, in which case the vectors \mathbf{u} and \mathbf{u+v} are called the end points of L.\,\!

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

 L = \{ \mathbf{u}+t\mathbf{v} \mid t\in(0,1)\}

for some vectors \mathbf{u}, \mathbf{v} \in V\,\! with  \mathbf{v} \neq \mathbf{0}.

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

[edit] Properties

[edit] See also

[edit] External links

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This article incorporates material from Line segment on PlanetMath, which is licensed under the GFDL.