LINE SEGMENT

In geometry, a 'line segment' is a part of a line that is bounded by two end points, which have a finite length, 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).

Contents

Definition

Properties

See also

External links

Definition

If $V,!$ 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
:$L\; =\; \{\; mathbf\{u\}+tmathbf\{v\}\; mid\; tin[0,1]\}$
for some vectors $mathbf\{u\},\; mathbf\{v\}\; in\; V,!$ with $mathbf\{v\}$
eq mathbf{0}, in which case the vectors $mathbf\{u\}$ and $mathbf\{u+v\}$ 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
:$L\; =\; \{\; mathbf\{u\}+tmathbf\{v\}\; mid\; tin(0,1)\}$
for some vectors $mathbf\{u\},\; mathbf\{v\}\; in\; V,!$ with $mathbf\{v\}$
eq mathbf{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 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.

See also

★ Interval (mathematics)

External links

★ Definiton of line segment With interactive animation

★ Copying a line segment with compass and straightedge

★ Dividing a line segment into N equal parts with compass and straightedge Animated demonstration
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