line segmentLineSegment2004-04-19 13:38:112006-08-20 03:24:34Definitionopen line segmentclosed line segment% this is the default PlanetMath preamble. as your knowledge
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\newcommand*{\abs}[1]{| #1 |}{\bf Definition}
Suppose $V$ is a vector space over $\sR$ or $\sC$, and $L$ is a subset of $V$.
Then $L$ is a \emph{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 \PMlinkname{closed}{Closed}
line segments. Then one defines a \emph{closed line segment}
as above,
and an \emph{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 $x \ne y$, then we denote by
$[x,y]$ the set connecting $x$ and $y$. This is , $\{\alpha x + (1-\alpha )y\ | 0 \le \alpha \le 1\}$. One can easily check that $[x,y]$ is a closed line segment.
\subsubsection*{Remarks}
\begin{itemize}
\item An alternative, equivalent, definition is as follows:
A (closed) line segment is a convex hull of two distinct points.
\item A line segment is connected, non-empty set.
\item 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.
\item More generally than above, the concept of a line segment can be
defined in an ordered geometry.
\end{itemize}