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).

Definition

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If $V\backslash ,\backslash !$ is a vector space over $\backslash mathbb\{R\}$ or $\backslash mathbb\{C\}$, and $L\backslash ,\backslash !$ is a subset of $V\backslash ,\backslash !.$ Then $L\backslash ,\backslash !$ is a line segment if $L\backslash ,\backslash !$ can be parametrized as

$L\; =\; \backslash \{\; a+tb\; \backslash mid\; t\backslash in*\backslash \}$

for some $a,\; b\backslash ,\backslash !$ in $V\backslash ,\backslash !$ with $b\backslash neq\; 0.$

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 $L\backslash ,\backslash !$ that can be parametrized as

$L\; =\; \backslash \{\; a+tb\; \backslash mid\; t\backslash in(0,1)\backslash \}$

for some $a,\; b\backslash ,$ in $V\backslash ,$ with $b\backslash neq\; 0.$

An alternative, equivalent, definition is as follows: A (closed) line segment is a convex hull of two distinct points.

Properties

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• 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 opened 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.

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See also

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• Interval (mathematics)

Elementary geometry | Linear algebra

segment | Jana (geometria) | segmento | 線分 | Odcinek | 线段