Definitions

# 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).

## Definition

If $V,!$ is a vector space over $mathbb\left\{R\right\}$ or $mathbb\left\{C\right\}$, and $L,!$ is a subset of $V,,!$ then $L,!$ is a line segment if $L,!$ can be parameterized as

$L = \left\{ mathbf\left\{u\right\}+tmathbf\left\{v\right\} mid tin\left[0,1\right]\right\}$

for some vectors $mathbf\left\{u\right\}, mathbf\left\{v\right\} in V,!$ with $mathbf\left\{v\right\} neq mathbf\left\{0\right\},$ in which case the vectors $mathbf\left\{u\right\}$ and $mathbf\left\{u+v\right\}$ 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

$L = \left\{ mathbf\left\{u\right\}+tmathbf\left\{v\right\} mid tin\left(0,1\right)\right\}$

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

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.