Line segment

In geometry, a line segment is a part of a straight line that is bounded by two distinct endpoints (its extreme points), and contains every point on the line that is between its endpoints. It is a special case of an arc, with zero curvature. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using an overline (vinculum) above the symbols for the two endpoints, such as in AB.

The infinite line containing a line segment is sometimes called the segment's supporting line.

Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (of that polygon or polyhedron) if they are adjacent vertices, or 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).

In real or complex vector spaces

If V is a vector space over ⁠

R

{\displaystyle \mathbb {R} }

⁠ or ⁠

C

,

{\displaystyle \mathbb {C} ,}

⁠ and L is a subset of V, then L is a line segment if L can be parameterized as

L

=

{

u

+

t

v

∣

t

∈

[

0

,

1

]

}

{\displaystyle L=\{\mathbf {u} +t\mathbf {v} \mid t\in [0,1]\}}

for some vectors

u

,

v

∈

V

{\displaystyle \mathbf {u} ,\mathbf {v} \in V}

where v is nonzero. The endpoints of L are then the vectors u and u + v.