Line segment

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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 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).

Definition

If [V\,\!] is a vector space over [\mathbb] or [\mathbb], and [L\,\!] is a subset of [V\,\!.] Then [L\,\!] is a line segment if [L\,\!] can be parametrized as

[ L = \]

for some [a, b\,\!] in [V\,\!] with [b\neq 0.]

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 = \]

for some [a, b\,] in [V\,] with [b\neq 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 opened 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.

This article incorporates material from on PlanetMath, which is licensed under the [Text of the GNU Free Documentation LicenseGFDL].

Line segment

Definition

Properties

See also