Line segment

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The geometric definition of a line segment

In geometry, a line segment is a part of a line that is bounded by two distinct 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

If is a vector space over or , and is a subset of then is a line segment if can be parametrized as

for some vectors with in which case the vectors and are called the end points of

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 that can be parametrized as

for some vectors with

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

Wikimedia Commons has media related to:

Line segment

Look up line segment in Wiktionary, the free dictionary.

• Definition 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.