Line segment

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Categories: Elementary geometry | Linear algebra

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).

1 Definition
2 Properties
3 See also
4 External links

Definition

If Failed to parse (Missing texvc executable; please see math/README to configure.): V\,\!

is a vector space over Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}
or  Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{C}

, and Failed to parse (Missing texvc executable; please see math/README to configure.): L\,\!

is a subset of Failed to parse (Missing texvc executable; please see math/README to configure.): V,\,\!
then Failed to parse (Missing texvc executable; please see math/README to configure.): L\,\!
is a line segment if Failed to parse (Missing texvc executable; please see math/README to configure.): L\,\!
can be parametrized as

Failed to parse (Missing texvc executable; please see math/README to configure.): L = \{ \mathbf{u}+t\mathbf{v} \mid t\in[0,1]\}

for some vectors Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{u}, \mathbf{v} \in V\,\!

with Failed to parse (Missing texvc executable; please see math/README to configure.):  \mathbf{v} \neq \mathbf{0},
in which case the vectors Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{u}
and Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{u+v}
are called the end points of Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): L\,\!

that can be parametrized as

Failed to parse (Missing texvc executable; please see math/README to configure.): L = \{ \mathbf{u}+t\mathbf{v} \mid t\in(0,1)\}

for some vectors Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{u}, \mathbf{v} \in V\,\!

with Failed to parse (Missing texvc executable; please see math/README to configure.):  \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 Failed to parse (Missing texvc executable; please see math/README to configure.): V

is a topological vector space, then a closed line segment is a closed set in Failed to parse (Missing texvc executable; please see math/README to configure.): V.
However, an open line segment is an open set in Failed to parse (Missing texvc executable; please see math/README to configure.): V
if and only if Failed to parse (Missing texvc executable; please see math/README to configure.): V
is one-dimensional.