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line segment (Definition)

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

$\displaystyle L = \{ a+tb \mid t\in[0,1]\}$
for some $ a,b$ in $ V$ with $ b\neq 0$.

Sometimes one needs to distinguish between open and closed 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

$\displaystyle L = \{ a+tb \mid t\in(0,1)\}$
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\ \vert 0 \le \alpha \le 1\}$. One can easily check that $ [x,y]$ is a closed line segment.

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"line segment" is owned by matte. [ full author list (3) ]
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See Also: interval, linear manifold, line in the plane

Also defines:  open line segment, closed line segment

Attachments:
endpoint (Definition) by Wkbj79

Cross-references: ordered geometry, open set, closed set, topological vector space, connected, points, convex hull, closed, equivalent, vectors, open, subset, vector space
There are 54 references to this entry.

This is version 9 of line segment, born on 2004-04-19, modified 2006-08-20.
Object id is 5783, canonical name is LineSegment.
Accessed 7679 times total.

Classification:
AMS MSC: 03-00 (Mathematical logic and foundations :: General reference works )
 51-00 (Geometry :: General reference works )
Pending Errata and Addenda
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