# line segment

Definition Suppose $V$ 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

 $L=\{a+tb\mid t\in[0,1]\}$

for some $a,b$ in $V$ with $b\neq 0$.

Sometimes one needs to distinguish between open and closed (http://planetmath.org/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

 $L=\{a+tb\mid t\in(0,1)\}$

for some $a,b$ in $V$ with $b\neq 0$.

If $x$ and $y$ are two vectors in $V$ and $x\neq y$, then we denote by $[x,y]$ the set connecting $x$ and $y$. This is , $\{\alpha x+(1-\alpha)y\ |0\leq\alpha\leq 1\}$. One can easily check that $[x,y]$ is a closed line segment.

## Remarks

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

• A line segment is 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.

 Title line segment Canonical name LineSegment Date of creation 2013-03-22 14:19:01 Last modified on 2013-03-22 14:19:01 Owner matte (1858) Last modified by matte (1858) Numerical id 12 Author matte (1858) Entry type Definition Classification msc 03-00 Classification msc 51-00 Related topic Interval Related topic LinearManifold Related topic LineInThePlane Related topic CircularSegment Defines open line segment Defines closed line segment