Segment (mathematics)

Segment (mathematics)
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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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1 Definition
2 Properties
3 In Proofs
4 See also
5 External links

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 parameterized 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.

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.