Line segment

In geometry, a line segment is a part of a straight line that is bounded by two distinct end 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 $AB$ .^[1]

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).

A line segment is a , non-empty set.

connected

If $V$ is a , 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.

topological vector space

More generally than above, the concept of a line segment can be defined in an .

ordered geometry

A pair of line segments can be any one of the following: , parallel, skew, or none of these. The last possibility is a way that line segments differ from lines: if two nonparallel lines are in the same Euclidean plane then they must cross each other, but that need not be true of segments.

intersecting

In proofs[edit]

In an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or defined in terms of an isometry of a line (used as a coordinate system).

Segments play an important role in other theories. For example, in a convex set, the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of convex sets, to the analysis of a line segment. The segment addition postulate can be used to add congruent segment or segments with equal lengths, and consequently substitute other segments into another statement to make segments congruent.

As a degenerate ellipse[edit]

A line segment can be viewed as a degenerate case of an ellipse, in which the semiminor axis goes to zero, the foci go to the endpoints, and the eccentricity goes to one. A standard definition of an ellipse is the set of points for which the sum of a point's distances to two foci is a constant; if this constant equals the distance between the foci, the line segment is the result. A complete orbit of this ellipse traverses the line segment twice. As a degenerate orbit, this is a radial elliptic trajectory.

Generalizations[edit]

Analogous to straight line segments above, one can also define arcs as segments of a curve.

In one-dimensional space, a ball is a line segment.

An oriented plane segment or bivector generalizes the directed line segment.

Chord (geometry)

Diameter

Radius

Polygonal chain

Interval (mathematics)

the algorithmic problem of finding intersecting pairs in a collection of line segments

Line segment intersection

The Foundations of Geometry. The Open Court Publishing Company 1950, p. 4

David Hilbert

"Line segment". MathWorld.

Weisstein, Eric W.

at PlanetMath

Line Segment

Copying a line segment with compass and straightedge

Animated demonstration

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

This article incorporates material from Line segment on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.