In geometry, a line segment is a part of a line that is bounded by two end points, 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).
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Definition
If <math>V\,\!</math> is a vector space over <math>\mathbb{R}</math> or <math>\mathbb{C}</math>, and <math>L\,\!</math> is a subset of <math>V,\,\!</math> then <math>L\,\!</math> is a line segment if <math>L\,\!</math> can be parametrized as
- <math> L = \{ \mathbf{u}+t\mathbf{v} \mid t\in[0,1]\}</math>
for some vectors <math>\mathbf{u}, \mathbf{v} \in V\,\!</math> with <math> \mathbf{v} \neq \mathbf{0},</math> in which case the vectors <math>\mathbf{u}</math> and <math>\mathbf{u+v}</math> are called the end points of <math>L.\,\!</math> Sometimes one needs to distinguish between "open" and "closed" line segments these also coinside with other line segments. Then one defines a closed line segment as above, and an open line segment as a subset <math>L\,\!</math> that can be parametrized as
- <math> L = \{ \mathbf{u}+t\mathbf{v} \mid t\in(0,1)\}</math>
for some vectors <math>\mathbf{u}, \mathbf{v} \in V\,\!</math> with <math> \mathbf{v} \neq \mathbf{0}.</math> 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 <math>V</math> is a topological vector space, then a closed line segment is a closed set in <math>V.</math> However, an open line segment is an open set in <math>V</math> if and only if <math>V</math> is one-dimensional.
- More generally than above, the concept of a line segment can be defined in an ordered geometry.
See also
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
This article incorporates material from Line segment on PlanetMath, which is licensed under the GFDL.
